Improved wave functions for
quantum Monte Carlo
Priyanka Seth
Corpus Christi College, Cambridge
Dissertation submitted for the degree of Doctor of Philosophy
at the University of Cambridge
August 2012
To mum, dad, ìÉìÉ and the memory of êÉ.
Preface
T
his dissertation describes work carried out between October 2008 and August 2012
in the Theory of Condensed Matter group at the Cavendish Laboratory, Cambridge
under the supervision of Prof. R. J. Needs. The following chapters are included in work
that has either been published or is to be published:
Chapter 3: P. Seth, P. Lo´pez Rı´os and R. J. Needs, “Quantum Monte Carlo study
of the first-row atoms and ions”, J. Chem. Phys. 134, 084105 (2011).
Chapter 4: P. Lo´pez R´ıos, P. Seth, N. D. Drummond and R. J. Needs, “Framework
for constructing generic Jastrow correlation factors”, Phys. Rev. E 86, 036703 (2012).
This dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration with others, except where specifically indicated
in the text. This dissertation has not been submitted in whole or in part for any other
degree or diploma at this or any other university. This dissertation does not exceed the
word limit of 60,000 words.
Priyanka Seth
Cambridge, August 2012
i
Abstract
Q
uantum Monte Carlo (QMC) methods can yield highly accurate energies for
correlated quantum systems. QMC calculations based on many-body wave func-
tions are considerably more accurate than density functional theory methods, and their
accuracy rivals that of the most sophisticated quantum chemistry methods. This the-
sis is concerned with the development of improved wave function forms and their use in
performing highly-accurate quantum Monte Carlo calculations.
All-electron variational and diffusion Monte Carlo (VMC and DMC) calculations are
performed for the first-row atoms and singly-positive ions. Over 98% of the correlation
energy is retrieved at the VMC level and over 99% at the DMC level for all the atoms
and ions. Their first ionization potentials are calculated within chemical accuracy. Scalar
relativistic corrections to the energies, mass-polarization terms, and one- and two-electron
expectation values are also evaluated. A form for the electron and intracule densities is
presented and fits to this form are performed.
Typical Jastrow factors used in quantum Monte Carlo calculations comprise electron-
electron, electron-nucleus and electron-electron-nucleus terms. A general Jastrow fac-
tor capable of correlating an arbitrary of number of electrons and nuclei, and including
anisotropy is outlined. Terms that depend on the relative orientation of electrons are
also introduced and applied. This Jastrow factor is applied to electron gases, atoms and
molecules and is found to give significant improvement at both VMC and DMC levels.
Similar generalizations to backflow transformations will allow useful additional vari-
ational freedom in the wave function. In particular, the use of different backflow func-
tions for different orbitals is expected to be important in systems where the orbitals are
qualitatively different. The modifications to the code necessary to accommodate orbital-
dependent backflow functions are described and some systems in which they are expected
to be important are suggested.
ii
Acknowledgements
F
irstly, I thank Richard Needs, to whom I am grateful for his eternal availability,
support and guidance over the course of my PhD. Muchas gracias to Pablo Lo´pez
R´ıos, who has been instrumental in transforming me into a nearly competent programmer
and a pedant when it comes to indentation. I thank Neil Drummond and John Trail for
helpful discussions. I am indebted to Tracey Ingham, Michael Rutter, David Taylor
and Helen Verrechia for keeping TCM running smoothly. This work would not have
been possible without financial support from the Cambridge Commonwealth Trust, TCM,
Corpus Christi College and the Cambridge Philosophical Society. The Cambridge HPC
Service provided computing resources.
My PhD experience was further enriched by my peers who provided education and
entertainment within and outside the lab. Amongst others, I would like to thank William
Belfield for introducing me to cryptics; Richard Brierley for his indispensable insight, both
scientific and otherwise; Danny Cole for helping diversify my sporting appetite; Anthony
Leung for bringing me down to earth; Felix Nissen for competitive tally collecting; Kayvan
Sadeghzadeh for providing balance; and Robert Lee and Alex Silver for infallibly executing
the duties of dream-team office mates. Alston Misquitta and Ioana Campean provided a
home away from home, complete with many a hot meal and endless mockery.
I would like to give a special mention to Mimi Hou and Sebastian Herbstreuth, who
have helped keep me sane over these few years. SB, CUlt and Punt provided the necessary
distraction in form of flying objects. Thanks also to the many others for whom I cannot
find the space to name individually.
I am grateful to Mike for being a rock on whose support I could always rely. I also
owe a gargantuan thank you to my family whose encouragement I have had behind me
throughout my PhD and who have tolerated my consequential absence from home.
iii
Contents
Preface i
Abstract ii
Acknowledgements iii
1 Introduction 1
1.1 Electronic structure calculations 1
1.1.1 Born-Oppenheimer Approximation 2
1.1.2 Variational principle 2
1.1.3 Electronic correlation 3
1.2 Common electronic structure methods 4
1.2.1 Hartree-Fock theory 4
1.2.2 Configuration interaction 5
1.2.3 Multi-configurational self-consistent field 6
1.2.4 Coupled cluster 7
1.2.5 Explicitly-correlated methods 7
1.2.5.1 Transcorrelated method 8
1.2.6 Density functional theory 8
1.2.7 Quantum Monte Carlo methods 10
1.2.8 Discussion 11
1.3 This thesis 12
2 Quantum Monte Carlo 13
2.1 Monte Carlo methods 13
2.1.1 The Metropolis algorithm 14
2.2 Variational Monte Carlo 15
2.3 Diffusion Monte Carlo 16
2.3.1 Imaginary-time evolution 16
2.3.2 The fermion sign problem 17
2.3.3 Importance sampling 19
iv
2.3.3.1 Drift-diffusion 20
2.3.3.2 Branching 21
2.3.4 Expectation values 22
2.4 The trial wave function 24
2.4.1 Characteristics of the wave function 24
2.4.2 The Slater determinant 25
2.4.3 The Jastrow factor 26
2.4.4 Backflow transformations 26
2.4.5 Multi-determinant expansions 27
2.4.6 Pairing wave functions 28
2.5 Errors, statistics and implementation 28
2.5.1 Serial correlation 29
2.5.2 Non-Gaussian distributions of estimates 30
2.5.3 Improving the efficiency of a QMC calculation 31
2.6 Optimization 33
2.6.1 Correlated sampling 34
2.6.2 Variance minimization 34
2.6.3 Energy minimization 36
3 Studies of the First-Row Atoms 38
3.1 Introduction 38
3.2 Trial wave functions 38
3.3 Optimization 40
3.4 Results and discussion 41
3.4.1 Atomic and ionic energies 41
3.4.2 Ionization potentials 46
3.4.3 Electron and electron-pair densities 49
3.4.4 Other expectation values 51
3.5 Conclusions 54
4 A Generic Jastrow Correlation Factor 57
4.1 Introduction 57
4.2 Construction of a generic Jastrow factor 58
4.2.1 Indexing of basis functions 60
4.2.2 Constraints 61
4.2.2.1 Symmetry constraints 61
4.2.2.2 Constraints at e–e and e–n coalescence points 62
4.2.2.3 Other constraints 62
4.3 Basis functions and terms 63
4.4 Results 66
v
4.4.1 Homogeneous electrons gases 66
4.4.1.1 One-dimensional homogeneous electron gas 66
4.4.1.2 Two-dimensional homogeneous electron gas 68
4.4.2 Be, B and O atoms 68
4.4.3 BeH, N2, H2O and H2 molecules 72
4.4.3.1 BeH molecule 72
4.4.3.2 N2 molecule 72
4.4.3.3 H2O molecule 76
4.4.3.4 H2 singlet 77
4.4.3.5 H2 triplet 80
4.4.4 Discussion of molecular results 87
4.4.5 Summary of results 91
4.5 Conclusions 91
5 Orbital-Dependent Backflow Transformations 94
5.1 Introduction 94
5.2 Implementation 96
5.2.1 Management of orbitals 96
5.2.2 Kinetic energy evaluation 97
5.3 Variational freedom 99
5.4 Systems of interest 101
5.5 Summary 102
6 Conclusions 103
References 104
vi
Chapter 1
Introduction
1.1 Electronic structure calculations
T
he behaviour of matter around us is determined by the behaviour of the par-
ticles that constitute it. In condensed matter systems, these elementary building
blocks are the nuclei that create the backbone of the material and the electrons that
bind the nuclei together. Ab initio electronic structure methods give an understanding
of the qualitative behaviour of a broad range of quantum systems and allow quantitative
predictions of their properties to be made without any prior knowledge of the system. Be-
sides being of tremendous importance for systems where obtaining experimental results is
difficult, theoretical methods also give a deeper insight into the physics of such systems.
By the first postulate of quantum mechanics, the wave function Ψ(R, t) contains all
the information specifying the state of the system. One only needs to solve the time-
dependent Schro¨dinger equation which, in atomic units, is
i
∂
∂t
Ψ(R, t) = HˆΨ(R, t), (1.1)
subject to symmetry constraints imposed by the nature of the particles, to understand
the structure of the material of interest. The 3N -dimensional vector R = (r1, r2, . . . , rN)
denotes the positions of the N particles. For systems comprising N electrons and M
nuclei, the non-relativistic Hamiltonian reads
Hˆ = −
N∑
i=1
∇2i
2
−
M∑
I=1
∇2I
2mI
+
N∑
i 0, and
ΘSµ(r) = g
S(r)θSµ(r) (4.7)
for µ > 0, where fP and gS are the e–e and e–n cut-off functions and φPν and θ
S
µ are
functions from a suitable basis set. This factorization allows an efficient implementation
of localized Jastrow factor terms.
Additionally, we allow expansion indices to take a value of zero with the special mean-
ing that ΦP0 (r) = Θ
S
0 (r) = 1 for all P , S, and r. Note that these zeroth-order functions do
not contain cut-off functions. This allows us to construct terms with specialized functional
forms, such as those involving dot products of vectorial quantities.
4.2.2 Constraints
Constraints on the parameters can be expressed in the form of a system of equations
involving the linear parameters and the basis function parameters. We restrict our analysis
to linear constraints on the linear parameters, and constraints that can be imposed on
the non-linear parameters contained in a basis function independently from the linear
parameters and from non-linear parameters in other basis functions.
Linear constraints on the linear parameters can be imposed using Gaussian elimina-
tion, as described in Ref. [84]. The matrix of coefficients may depend on the non-linear
parameters in the basis functions, if present, and the linear system is usually underde-
termined, resulting in a subset of the parameters being determined by the values of the
remaining parameters, which can be optimized directly.
4.2.2.1 Symmetry constraints
Equation (4.4) imposes the condition that Jn,m(i, I) must not depend on the specific
ordering of the electrons and nuclei listed in i and I. The linear parameters of the
Jastrow factor must exhibit a symmetry that implies that a parameter with a given
set of superindices {P(i), S(i, I)} is determined by another parameter with a permuted set
of superindices {P′(i), S′(i, I)}. This redundancy is removed by considering only one of
the possible permutations of {P(i), S(i, I)}, or the signature. These symmetry constraints
amount to equalities between pairs of parameters and must always be imposed, otherwise
the trial wave function is unphysical and calculations give erroneous results.
61
4.2 CONSTRUCTION OF A GENERIC JASTROW FACTOR
4.2.2.2 Constraints at e–e and e–n coalescence points
The Coulomb potential energy diverges when the positions of two electrons or an electron
and a nucleus coincide. However, the local energy of an eigenstate of the Hamiltonian,
including the exact ground-state wave function, is finite and constant throughout config-
uration space. Divergences in the local energy are therefore not a feature of the exact
wave function and can lead to poor statistics in QMC calculations; hence it is impor-
tant to avoid them. The kinetic energy must diverge to cancel out the potential energy
and keep the local energy finite, which is achieved by demanding that the wave function
obeys the Kato cusp conditions [125]. For any two charged particles i and j in a two- or
three-dimensional system interacting via the Couloumb potential, these are(
1
Ψ
∂Ψˆ
∂rij
)
rij→0
=
2qiqjµij
d± 1 , (4.8)
where Ψˆ denotes the spherical average of Ψ, q represents charge, µij = mimj/(mi+mj) is
the reduced mass, m represents mass, d is the dimensionality, and the positive sign in the
denominator is for indistinguishable particles and the negative sign is for distinguishable
particles. Fixed nuclei are regarded as having an infinite mass.
As typical forms of ΨD explicitly depend only on e–n distances, it is common practice
to impose the e–n cusp conditions on ΨD and the e–e cusp conditions on the Jastrow
factor. Our implementation allows the option of applying both types of cusp conditions
to the Jastrow factor, which gives flexibility in the choice of ΨD and its properties. In
particular, we impose the cusp conditions on a single Jastrow factor term, and constrain
all other terms in the Jastrow factor so that their contribution to the local kinetic energy
is finite at e–e and e–n coalescence points. For non-divergent interaction potentials, such
as most pseudopotentials, we simply require that the kinetic energy remains finite at
coalescence points. Our implementation is also capable of not applying any constraints
at e–e and e–n coalescence points since this is advantageous in some cases [128, 129], as
discussed in Sec. 4.4.3.5.
Imposing that the kinetic energy be finite at coalescence points is non-trivial if the
Jastrow factor contains anisotropic functions. As two particles coalesce, ∇J and ∇2J
must remain finite, and this gives rise to further constraints that must be satisfied.
4.2.2.3 Other constraints
It is possible to construct terms containing dot products by using appropriate constraints.
For example, consider the basis functions Θ1(r) = x, Θ2(r) = y, and Θ3(r) = z. In an
e–n–n term we can restrict the indices so that µ takes only the values (1 1), (2 2), (3 3),
so that the contribution of electron i and nuclei I and J is riI ·riJ , provided we also apply
a linear constraint that equates the three non-zero linear coefficients.
62
4.3 BASIS FUNCTIONS AND TERMS
It is also possible to introduce Boys-Handy-style indexing [13], where the sum of all
e–e and e–n indices is restricted to be less than or equal to some fixed integer l.
4.3 Basis functions and terms
We employ a condensed notation to refer to Jastrow terms that use certain basis functions,
cut-off functions and constraints. Each term is represented by a single capital letter, with
n and m as subindices. Any other relevant information is given as a superindex. Table 4.1
summarizes the notation for the terms we have introduced.
Possibly the simplest basis set is the natural powers,
Nν(r) = r
ν−1, (4.9)
as used in the DTN Jastrow factor for the localized u, χ, and f terms [84]. These functions
need to be cut off at some radius L, for which purpose the DTN Jastrow factor uses the
polynomial cut-off function
D(r) = (r − L)CΘ(L− r), (4.10)
where L is an optimizable parameter, C is a positive integer, and Θ(r) is the Heaviside
step function. We also use a slightly different version of this cut-off function,
P (r) = (1− r/L)CΘ(L− r), (4.11)
which should be numerically superior to D(r).
For simple Jastrow terms we use the natural power basis functions Nν and the poly-
nomial cut-off functions P or D. We refer to these terms as Nn,m. N2,0, N1,1, and N2,1
are the equivalent of the DTN u, χ, and f terms, respectively. In the N2,1 term, and in
any term where more than one electron and one or more nuclei are involved, we choose
not to apply e–e cut-off functions, relying instead on the e–n cut-offs to fulfill this role.
Additional Nn,m terms used here that were not part of the DTN Jastrow factor are N1,2,
N3,0, N1,3, N2,2, N3,1, and N4,0. In Nn,m we typically use a truncation order in the cut-off
function of C = 3 to ensures that the local energy is continuous.
A particular variant of P (r) is the anisotropic cut-off function
A(r) = (1− r/L)CΘ(L− r)
∑
i
ci
d∏
β
[
r · uˆβ
r
]p(i)β
, (4.12)
where L is an optimizable parameter, C is a positive integer, d is the dimensionality of the
system, uˆβ are unit vectors along d orthogonal directions, ci are real-valued constants, and
63
4.3 BASIS FUNCTIONS AND TERMS
p
(i)
β are integer exponents, which are constrained so that
∑d
β p
(i)
β is the same for all values
of i. This cut-off function is simply the product of an isotropic cut-off function and a
spherical harmonic. For example, with c1 = 3, c2 = −1, p(1) = (2 1 1), and p(2) = (0 3 1),
and the vectors pointing along the Cartesian axes, we obtain
A(r) = (1− r/L)CΘ(L− r)
[
(3x2 − y2)yz
r4
]
, (4.13)
which is proportional to a real spherical harmonic with l = 4. The advantage of describing
anisotropy in the cut-off function rather than in the basis functions is that the common
spherical harmonic can be factorized out of the sum over expansion indices, which reduces
the computational cost. We allow different orientations to be used for different e–e or e–n
dependency indices, which is useful to adapt the functional form to, e.g., the geometry of
a molecule.
We use As.h.n,m to refer to the anisotropic variant of Nn,m. This term consists of natural
power basis functions Nν and the anisotropic cut-off function A, and “s.h.” is a placeholder
for the description of the spherical harmonic. For example, for the highly anisotropic N2
molecule we use terms such as Az1,1, A
z2
1,1, A
z
2,1, and A
z2
2,1.
An alternative to the natural-power basis in finite systems is a basis of powers of
fractions which tend to a constant as r →∞, and therefore do not need to be cut off. We
define the basis
Fν(r) =
(
r
rb + a
)ν−1
, (4.14)
where a and b are real-valued optimizable parameters. Similar basis sets with b = 1
have been used in the literature, often in conjunction with Boys-Handy-style indexing
[13, 130, 131, 128], and this basis was used in Chapter 3 with an early implementation of
the Jastrow factor presented here.
The Fν basis functions are used in terms Fn,m, or F
b=1
n,m when we force b = 1 in the
basis functions. In some systems it is useful to apply Boys-Handy-style indexing to F b=1n,m ,
and we refer to the resulting term as Bn,m.
In extended systems it is important to use a basis that is consistent with the geometry
of the simulation cell and has the periodicity of the system, such as a cosine basis,
Cν(r) =
∑
G∈ν-th star
cos (G · r) , (4.15)
where the G vectors are arranged in stars defined by the cell geometry.
Terms denoted by Cn,m use of the cosine basis functions Cν . We choose expansion
orders so that at least as many G vectors as electrons in each spin channel are included
in the expansion. These terms capture correlation in the corners of the simulation cell
beyond the cut-off radius of Nn,m terms. They are also important in describing strongly
64
4.3 BASIS FUNCTIONS AND TERMS
anisotropic materials. However, the long-range nature of these terms makes them more
computationally expensive to evaluate.
C2,0 and C1,1 correspond to the DTN p and q terms, respectively. While these terms
are computationally expensive to evaluate as they are not cut off at any distance, they
are also important in describing strongly anisotropic materials.
A suitable basis set for building specialized terms containing dot products is
Vν(r) = r
int[(ν−1)/d] r · uˆmod(ν−1,d)+1
r
, (4.16)
where d is the dimensionality of the system and uˆβ are the d unit vectors parallel to the
Cartesian axes. A term constructed using these functions with appropriate constraints
would consist of dot products between two vectors multiplied by a natural-power expan-
sion in their moduli.
To test the flexibility of our implementation we have designed an e–e–n–n Jastrow
term for describing the correlations associated with van der Waals interactions, which
we call V2,2. This term is capable of distinguishing between configurations where the
electron-nucleus relative position vectors riI and rjJ are parallel from those where they
are anti-parallel. Introducing a dot product achieves this effect, and V2,2 has the following
functional form,
V2,2 =
1
2
N∑
i 6=j
M∑
I 6=J
P (riI)P (rjJ)
p∑
νij
q∑
µiI ,µjJ
λνijµiIµiJ
×Nνij(rij)NµiI (riI)NµjJ (rjJ)riI · rjJ . (4.17)
We require basis functions to be scalars in our Jastrow factor, so the dot product is
separated into its components. Hence, we construct the V2,2 term using Vν for the e–n
basis with P as the e–n cut-off functions, and Nν for the e–e basis. We allow e–n indices
to be zero.
table 4.1: Notation for Jastrow terms correlating n electrons and m nuclei using different
basis functions.
Name Basis set Cut-off function Special constraints
Nn,m Natural powers Polynomial None
Fn,m Powers of r/(r
b + a) None None
Bn,m Powers of r/(r + a) None Boys-Handy-style indexing
As.h.n,m Natural powers Anisotropic polynomial None
Cn,m Cosines None None
Vn,m Natural powers times unit vectors Polynomial Dot product
65
4.4 RESULTS
4.4 Results
We have used a variety of methods to optimize our Jastrow factors, namely variance
minimization, minimization of the mean absolute deviation of the local energy with respect
to the median energy, and linear least-squares VMC energy minimization [81, 82]. All of
our final wave functions are energy-minimized except where otherwise stated. Starting
with the Hartree-Fock wave function, we progressively introduce Jastrow terms and re-
optimize all of the parameters simultaneously. Optimizing the Jastrow factor term-by-
term is unnecessary in practical applications, but here it allows us to understand the
importance of the different terms. We refer to the total number of optimizable parameters
in the wave function as Np.
To measure the quality of the trial wave function Ψ, we again use the fraction of the
correlation energy retrieved in a VMC calculation with a given trial wave function Ψ,
fCE[Ψ] =
EHF − EVMC[Ψ]
EHF − Eexact . (4.18)
We refer to the difference between the DMC and HF energies as the DMC correlation
energy, EHF − EDMC[Ψ]. The fraction of the DMC correlation energy retrieved in VMC,
fDCE[Ψ] =
EHF − EVMC[Ψ]
EHF − EDMC[Ψ] , (4.19)
measures the quality of the Jastrow factor, since a perfect Jastrow factor would make
the VMC and DMC energies coincide3. We define the fraction of the remaining DMC
correlation energy recovered by a wave function Ψ2 with respect to another Ψ1 as
EVMC[Ψ1]− EVMC[Ψ2]
EVMC[Ψ1]− EDMC[Ψ2] . (4.20)
As VMC variance tends to its lower bound of zero as Ψ tends to an eigenstate of
the Hamiltonian, the variance is also a measure of the overall quality of the trial wave
function.
4.4.1 Homogeneous electrons gases
4.4.1.1 One-dimensional homogeneous electron gas
We have studied a 1D homogeneous electron gas (HEG) of density parameter rs = 5 a.u.
consisting of 19 electrons subject to periodic boundary conditions using a single Slater
determinant of plane-wave orbitals. The ground-state energy of an infinitely thin 1D HEG
in which electrons interact by the Coulomb potential is independent of the magnetic state,
3In general, attaining the DMC limit in VMC would require non-analyticities in the Jastrow factor.
Nonetheless this theoretical limit is useful for assessing the performance of a Jastrow factor.
66
4.4 RESULTS
and hence we have chosen all the electrons to have the same spin. This system is unusual
in that the nodal surface of the trial function is exact, and therefore DMC gives the exact
ground-state energy, which we have estimated to be −0.2040834(3) a.u. per electron.
Excellent results were reported for this system in Refs. [133, 134] using wave functions
with e–e backflow transformations [124, 69] which preserve the (exact) nodal surface of the
Slater determinant. Our VMC results for different Jastrow factors are given in Table 4.2.
table 4.2: Energies (E) and VMC variances (V ) of the 1D HEG at rs = 5 a.u. using
different Jastrow factors. The use of backflow is indicated by “(BF)”.
Np E (a.u. per electron) V (a.u.) fDCE (%)
HF −0.191653064 0
N2,0 9 −0.204076(1) 0.0000654(7) 99.941(8)
N2,0+C2,0 18 −0.2040824(7) 0.0000168(3) 99.992(6)
N2,0+C2,0+N3,0 45 −0.2040831(2) 0.00000171(3) 99.998(3)
N2,0+C2,0+C3,0 52 −0.2040832(6) 0.0000127(3) 99.998(5)
N2,0+C2,0+N3,0+C3,0 79 −0.2040833(2) 0.00000105(3) 99.999(3)
N2,0 (BF) 18 −0.2040816(5) 0.00000809(6) 99.986(5)
N2,0+C2,0 (BF) 27 −0.2040833(2) 0.00000104(3) 99.999(3)
N2,0+C2,0+N3,0 (BF) 54 −0.2040832(1) 0.00000055(2) 99.998(3)
N2,0+C2,0+C3,0 (BF) 61 −0.20408310(7) 0.00000020(1) 99.998(3)
N2,0+C2,0+N3,0+C3,0 (BF) 88 −0.20408310(7) 0.00000020(1) 99.998(3)
DMC −0.2040834(3) 100.000(4)
table 4.3: e–e expansion orders (p) used for the different Jastrow terms in the 1D HEG.
N2,0 N3,0 C2,0 C3,0
p 9 5 10 5
We have investigated the improvement in VMC results when various terms are added
to an e–e Jastrow factor J = N2,0 +C2,0, both with and without backflow transformations.
In the absence of backflow, we find that including N3,0, C3,0, or N3,0 + C3,0 improves the
VMC energy, while the subsequent addition of C4,0 yields no further gain. We observe
a ten-fold reduction in the variance upon addition of N3,0 to J = N2,0 + C2,0. The C3,0
term does not duplicate the N3,0 term, and they combine to give a further reduction in
variance.
VMC gives an almost exact energy with backflow and J = N2,0 + C2,0, and therefore
no further reduction is possible by including more Jastrow terms. However, the addition
of N3,0 + C3,0 reduces the VMC variance by a factor of five, giving a variance that is an
order of magnitude smaller than that reported in Ref. [133] for a similar calculation.
The energy of a SJ wave function with a J = N2,0+C2,0+N3,0 containing 45 optimizable
parameters is within error bars of the exact (DMC) energy, while a SJB wave function
67
4.4 RESULTS
with only J = N2,0 +C2,0 and a total of 27 optimizable parameters is required to achieve
this. Backflow transformations introduce the variational freedom more compactly than
the N3,0 term.
4.4.1.2 Two-dimensional homogeneous electron gas
We have studied a paramagnetic 2D HEG with 42 electrons per simulation cell at rs =
35 a.u., which lies close to the ferromagnetic Wigner crystallization density predicted by
Drummond and Needs [72]. Kwon et al. [124] found that three-electron correlations are
important at low densities, and that the effect of a three-electron Jastrow factor on the
VMC energy is comparable to that of backflow. At higher densities, the effects of velocity-
dependent backflow transformations become more dominant. This makes low densities
appealing for testing higher-rank Jastrow terms. The VMC energy and variance obtained
using different Jastrow factors with and without backflow is plotted in Fig. 4.1 and the
results are given in Table 4.2.
The addition of an N3,0 term to J = N2,0 recovers 81% of the remaining DMC cor-
relation energy without backflow and 49% with backflow. The C2,0 term further reduces
both the VMC energy and variance. The use of a C3,0 term recovers 10% of the remaining
DMC correlation energy when added to J = N2,0 +C2,0, but it was not used further since
the lack of a cut-off function makes calculations with C3,0 too costly for the little benefit
it provides.
We have also performed DMC calculations using two different Jastrow factors in the
presence of backflow in order to quantify the indirect effect of the quality of the Jastrow
factor on the quality of the nodes of the wave function. We obtain a DMC energy of
−0.0277072(1) a.u. per electron for J = N2,0, and a lower energy of −0.0277087(1) a.u.
per electron for J = N2,0 +N3,0 +C2,0. This supports the idea that a better Jastrow factor
allows the backflow transformation to shift its focus from the “bulk” of the wave function
to its nodes, thus improving the DMC energy.
4.4.2 Be, B and O atoms
While excellent descriptions of these atoms can be obtained within VMC and DMC using
multi-determinant wave functions with backflow correlations [132, 85], we have used single-
determinant wave functions since we are only interested in the effects of the Jastrow factor.
The decrease in quality of the Jastrow factor for heavier atoms can be attributed to the
increase in inhomogeneity as Z increases. The higher-order terms are expected to therefore
give improve the wave function. We have studied the ground states of the Be, B, and
O atoms, corresponding to 1S, 2P, and 3P electronic configurations, respectively. The
atsp2k code [83] was used to generate numerical single-electron HF orbitals tabulated
on a radial grid. We have investigated the use of Jastrow factors with up to four-body
68
4.4 RESULTS
t
a
b
l
e
4.
4:
E
n
er
gi
es
(E
)
an
d
V
M
C
va
ri
an
ce
s
(V
)
of
th
e
2D
H
E
G
at
r s
=
35
a.
u
.
u
si
n
g
d
iff
er
en
t
J
as
tr
ow
fa
ct
or
s.
T
h
e
u
se
of
b
ac
k
fl
ow
is
in
d
ic
at
ed
b
y
“(
B
F
)”
.
N
p
E
(a
.u
.
p
er
el
ec
tr
on
)
V
(a
.u
.)
f D
C
E
(%
)
f D
C
E
(B
F
)
(%
)
H
F
−0
.0
17
12
17
92
0
0
N
2
,0
18
−0
.0
27
53
41
(8
)
0.
00
00
62
0(
5)
98
.7
6(
1)
N
2
,0
+
C
2
,0
24
−0
.0
27
54
00
(8
)
0.
00
00
59
3(
6)
98
.8
2(
1)
N
2
,0
+
N
3
,0
57
−0
.0
27
60
50
(6
)
0.
00
00
37
8(
4)
99
.4
3(
1)
N
2
,0
+
C
2
,0
+
N
3
,0
65
−0
.0
27
61
12
(6
)
0.
00
00
33
9(
4)
99
.4
9(
1)
N
2
,0
+
C
2
,0
+
C
3
,0
13
2
−0
.0
27
55
20
(7
)
0.
00
00
54
6(
6)
98
.9
3(
1)
N
2
,0
(B
F
)
35
−0
.0
27
60
09
(7
)
0.
00
00
46
2(
4)
98
.9
82
(7
)
N
2
,0
+
C
2
,0
(B
F
)
43
−0
.0
27
60
46
(7
)
0.
00
00
43
6(
5)
99
.0
17
(7
)
N
2
,0
+
N
3
,0
(B
F
)
76
−0
.0
27
65
41
(5
)
0.
00
00
26
9(
3)
99
.4
84
(5
)
N
2
,0
+
C
2
,0
+
N
3
,0
(B
F
)
84
−0
.0
27
66
14
(5
)
0.
00
00
23
2(
3)
99
.5
53
(5
)
D
M
C
−0
.0
27
66
49
(9
)
10
0.
00
(1
)
99
.5
86
(9
)
D
M
C
fo
r
N
2
,0
(B
F
)
−0
.0
27
70
72
(1
)
99
.9
86
(1
)
D
M
C
fo
r
N
2
,0
+
C
2
,0
+
N
3
,0
(B
F
)
−0
.0
27
70
87
(1
)
10
0.
00
0(
1)
69
4.4 RESULTS
0 2×10-5 4×10-5 6×10-5
VMC variance (a.u.)
-0.0277
-0.0276
-0.0275
E V
M
C
(a.
u.)
VMC
VMC (BF)
DMC
DMC (BF)
N2,0
N2,0
N2,0+C2,0
N2,0+C2,0
N2,0+C2,0+C3,0
N2,0+N3,0
N2,0+C2,0+N3,0
N2,0+N3,0
N2,0+C2,0+N3,0
figure 4.1: VMC energies against the VMC variance for the 2D HEG at rs = 35 a.u.
using different Jastrow factors, along with the DMC energies for reference. The error bars
are smaller than the size of the symbols, and “(BF)” indicates the use of backflow.
table 4.5: e–e expansion orders (p) used for the different Jastrow terms in the 2D HEG.
N2,0 N3,0 C2,0 C3,0
p 9 4 5 3
70
4.4 RESULTS
terms, but we have not used backflow for these systems. The energies of Chakravorty et
al. [88] have been used as “exact” reference values. Our VMC results for Be, B and O are
given in Table 4.6.
table 4.6: Energies (E) and VMC variances (V ) for the Be, B and O atoms using
different Jastrow factors.
Np E (a.u.) V (a.u.) fCE(%) fDCE(%)
Be atom
HF −14.573023 0 0
F2,0+F1,1+F2,1 103 −14.65062(7) 0.0445(5) 82.26(7) 92.22(9)
F2,0+F1,1+F2,1+F3,0 160 −14.6512(1) 0.0470(3) 82.9(1) 92.9(1)
F2,0+F1,1+F2,1+F3,1 170 −14.6522(1) 0.051(1) 83.9(1) 94.0(1)
VMC from Ref. [85] −14.6311(1) 61.6(1) 69.0(1)
VMC from Ref. [135] −14.64972(5) 81.30(5) 91.15(7)
DMC −14.65717(4) 89.20(4) 100.00(7)
Exact from Ref. [88] −14.66736 100
B atom
HF −24.529061 0 0
F2,0+F1,1+F2,1 103 −24.6299(1) 0.093(1) 80.77(8) 90.9(1)
F2,0+F1,1+F2,1+F3,0 185 −24.6302(1) 0.0960(5) 81.01(8) 91.1(1)
F2,0+F1,1+F2,1+F3,1 195 −24.6309(2) 0.0973(6) 81.6(2) 91.8(2)
VMC from Ref. [85] −24.6056(2) 61.3(2) 69.0(2)
VMC from Ref. [135] −24.62936(5) 80.34(4) 90.39(7)
DMC −24.64002(6) 88.87(5) 100.00(8)
Exact from Ref. [88] −24.65391 100
O atom
HF −74.809398 0 0
F2,0+F1,1+F2,1 103 −75.0341(2) 0.550(2) 87.13(8) 92.97(9)
F2,0+F1,1+F2,1+F3,0 185 −75.0368(4) 0.577(2) 88.2(2) 94.1(2)
F2,0+F1,1+F2,1+F3,1 195 −75.0381(3) 0.498(2) 88.7(1) 94.6(1)
VMC from Ref. [85] −75.0233(3) 82.9(1) 88.5(1)
VMC from Ref. [135] −75.0352(1) 87.55(4) 93.42(6)
DMC −75.0511(1) 93.72(4) 100.00(6)
Exact from Ref. [88] −75.0673 100
table 4.7: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the Be, B, and O atoms.
F2,0 F1,1 F2,1 F3,1
p 9 – 5 3
q – 9 5 3
We obtain lower single-determinant VMC energies for the Be, B, and O atoms with J =
F2,0 +F1,1 +F2,1 than reported in Refs. [85, 135]. We obtain further small improvements in
the VMC energies by including either F3,0 or F3,1 Jastrow terms, but their combination,
71
4.4 RESULTS
F3,0 + F3,1, is not found to be advantageous over using the terms individually. This
indicates that F3,0 and F3,1, the latter of which provides a slightly lower VMC energy
than the former, have nearly the same effect in these atoms. These three-electron terms
should be particularly useful in describing correlations involving electrons in the atomic
core region. We expect F3,1 to be more useful than F3,0 in molecules and solids because it
should be able to adapt to the different length scales in these systems, whereas F3,0 offers
a homogeneous description of three-electron correlations. We have investigated the effect
of adding a F4,1 term in Be and O, but it does not reduce the VMC energy or variance
when added to J = F2,0 + F1,1 + F2,1 + F3,1.
Our best VMC energies of −14.6522(1) a.u., −24.6309(2) a.u., and −75.0381(3) a.u.
for Be, B and O respectively correspond to fractions of the DMC correlation energy of
94.0(1)%, 91.8(1)%, and 94.6(1)%.
4.4.3 BeH, N2, H2O and H2 molecules
The BeH, N2 and H2O molecules are strongly inhomogeneous and anisotropic systems.
We have used basis sets of moderate quality for the single-electron orbitals of BeH and
N2 in order to investigate the extent to which the Jastrow factor can compensate for the
deficiencies of the basis sets, especially via one-electron terms N1,m. For H2O and the H2
triplet we have used very good basis sets. We have also tested anisotropic Jastrow factors
in N2, and a van der Waals-like Jastrow factor for H2.
4.4.3.1 BeH molecule
We have studied the all-electron BeH molecule in the 2Σ+ ground state configuration
at a bond length of 2.535 a.u. [136]. We have used a single-determinant wave function
containing Slater-type orbitals generated with the adf package [137], with which we
obtain a reference DMC energy of −15.24603(4) a.u. Our results are given in Table 4.8.
The addition of N1,2 to J = N2,0 + N1,1 + N2,1 recovers 11% of the remaining DMC
correlation energy. We find no significant gain from adding either an N2,2 term or an N3,1
term to J = N2,0 + N1,1 + N2,1 + N1,2, possibly due to the large number of parameters
that needed to be optimized.
4.4.3.2 N2 molecule
We have studied the 1Σ+g ground state of the N2 molecule at the experimental bond length
of 2.074 a.u. [136] HF orbitals were generated in a Slater-type basis using the adf package
[137]. Our VMC results for different Jastrow factors are given in Table 4.10 along with
relevant reference energies.
Adding an N1,2 term to J = N2,0 + N1,1 + N2,1 recovers 33% of the remaining DMC
correlation energy and leads to a significant reduction in the VMC variance. The sub-
72
4.4 RESULTS
t
a
b
l
e
4.
8:
E
n
er
gi
es
(E
)
an
d
V
M
C
va
ri
an
ce
s
(V
)
fo
r
th
e
B
eH
m
ol
ec
u
le
u
si
n
g
d
iff
er
en
t
J
as
tr
ow
fa
ct
or
s.
W
e
h
av
e
u
se
d
a
b
on
d
le
n
gt
h
of
r B
eH
=
2.
53
5
a.
u
.
[1
36
].
N
p
E
(a
.u
.)
V
(a
.u
.)
f C
E
(%
)
f D
C
E
(%
)
U
H
F
li
m
it
fr
om
[1
38
]a
−1
5.
15
36
0
0
U
H
F
(a
d
f
)
−1
5.
15
35
−0
.1
05
7
−0
.1
08
2
N
2
,0
27
−1
5.
17
75
(5
)
0.
56
81
(7
)
25
.3
(5
)
25
.9
(5
)
N
2
,0
+
N
1
,1
63
−1
5.
21
99
(2
)
0.
24
04
(2
)
70
.1
(2
)
71
.7
(2
)
N
2
,0
+
N
1
,1
+
N
1
,2
84
−1
5.
22
35
(2
)
0.
23
43
(3
)
73
.9
(2
)
75
.6
(2
)
N
2
,0
+
N
1
,1
+
N
2
,1
25
7
−1
5.
24
05
4(
8)
0.
04
17
(3
)
91
.9
0(
8)
94
.1
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
27
8
−1
5.
24
11
6(
7)
0.
03
92
(1
)
92
.5
6(
7)
94
.7
3(
9)
V
M
C
fr
om
R
ef
.
[1
39
]a
−1
5.
21
2(
1)
62
(1
)
63
(1
)
V
M
C
fr
om
R
ef
.
[1
40
]
−1
5.
22
8(
1)
79
(1
)
80
(1
)
D
M
C
−1
5.
24
60
3(
4)
97
.7
1(
4)
10
0.
00
(6
)
E
x
ac
t
fr
om
R
ef
s.
[1
38
]a
an
d
[1
36
]
−1
5.
24
82
10
0
a
R
ef
.
[1
38
]
u
se
d
r B
eH
=
2.
53
7
a.
u
.
an
d
R
ef
.
[1
39
]
u
se
d
r B
eH
=
2.
53
8
a.
u
.
W
e
d
o
n
ot
ex
p
ec
t
th
at
th
es
e
sm
al
l
d
iff
er
en
ce
s
in
b
on
d
le
n
gt
h
w
il
l
aff
ec
t
th
e
co
m
p
ar
is
on
b
et
w
ee
n
en
er
gi
es
si
gn
ifi
ca
n
tl
y.
73
4.4 RESULTS
table 4.9: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the BeH molecule.
N2,0 N1,1 N1,2 N2,1
p 9 – – 4
q – 9 4 4
sequent addition of N2,2 provides a reduction in the VMC energy of 13% of the re-
maining DMC correlation energy. We have tested adding N3,0, N3,1, and N4,0 terms
to J = N2,0 +N1,1 +N2,1 +N2,2, but neither of these yield any improvements in the VMC
energy.
The anisotropy of this system is expected to be captured by terms containing e–n
functions that treat the bond as a special direction. We have aligned the z-axis of our
reference frame along the N–N bond in our calculations, and Az1,1 is then the simplest
explicitly anisotropic term that reflects the geometry of the system. The Ax1,1 and A
y
1,1
terms must be zero by symmetry and we have therefore not used them. There are five
spherical harmonics with l = 2, which are respectively proportional to xy, xz, yz, x2−y2,
and −x2 − y2 + 2z2. We find that only the last one of these, which we refer to as z2, has
a significant effect on the VMC energy.
The VMC energy with J = N2,0 + N1,1 + N2,1 + A
z
1,1 is within statistical uncertainty
of that with J = N2,0 +N1,1 +N2,1 +N1,2, but the former Jastrow factor contains about
a third fewer parameters than the latter. The combination of the N1,2 and A
z
1,1 terms
into J = N2,0 + N1,1 + N2,1 + N1,2 + A
z
1,1 does not improve the VMC energy compared
with the other two Jastrow factors. These results suggest that the terms N1,2 and A
z
1,1
play similar roles in the wave function, which we find reasonable since N1,2, although
constructed from isotropic basis functions, contains the right variables to capture the
symmetry of the molecule in much the same way as Az1,1 does. We have plotted the A
z
1,1
term for J = N2,0 +N1,1 +N2,1 +A
z
1,1 and the N1,2 term for J = N2,0 +N1,1 +N2,1 +N1,2 in
Fig. 4.2, where the similarity between the terms can be seen. The value of the N1,2 term
is roughly the same as that of Az1,1 offset by a positive amount, and this shift is likely to
be compensated for by the other Jastrow factor terms. Both terms increase the value of
the wave function in the outer region of the molecule with respect to that in the bond
region.
We have added different combinations of anisotropic terms to J = N2,0 +N1,1 +N2,1.
The e–e–n Az2,1 term retrieves less correlation energy than the e–n A
z
1,1 term. The A
z
1,2
term does not improve the N2,0+N1,1+N1,2 Jastrow factor and it was not considered
further. Combining terms with spherical harmonics of l = 1 and l = 2 improves the VMC
energy significantly with respect to using l = 1 only. The anisotropic Jastrow factor
J = N2,0 +N1,1 +N2,1 +A
z
1,1 +A
z2
1,1 +A
z
2,1 +A
z2
2,1, which contains up to e–e–n correlations
and has 191 optimizable parameters, recovers 93.3(1)% of the DMC correlation energy.
74
4.4 RESULTS
t
a
b
l
e
4.
10
:
V
M
C
en
er
gi
es
(E
)
an
d
va
ri
an
ce
s
(V
)
fo
r
th
e
N
2
m
ol
ec
u
le
u
si
n
g
d
iff
er
en
t
J
as
tr
ow
fa
ct
or
s,
in
cl
u
d
in
g
ex
p
li
ci
tl
y
an
is
ot
ro
p
ic
te
rm
s.
W
e
h
av
e
u
se
d
a
b
on
d
le
n
gt
h
of
r N
N
=
2.
07
4
a.
u
.
[1
36
].
N
p
E
(a
.u
.)
V
(a
.u
.)
f C
E
(%
)
f D
C
E
(%
)
H
F
li
m
it
fr
om
R
ef
.
[1
38
]
−1
08
.9
92
9
0
0
H
F
(a
d
f
)
−1
08
.9
91
7
−0
.2
18
5
−0
.2
33
9(
3)
N
2
,0
18
−1
09
.1
02
(1
)
5.
27
5(
4)
19
.9
(2
)
21
.3
(2
)
N
2
,0
+
N
1
,1
27
−1
09
.3
73
9(
6)
3.
68
1(
3)
69
.4
(1
)
74
.3
(2
)
N
2
,0
+
N
1
,1
+
N
1
,2
49
−1
09
.3
79
6(
6)
3.
59
5(
2)
70
.4
(1
)
75
.4
(2
)
N
2
,0
+
N
1
,1
+
N
2
,1
80
−1
09
.4
44
1(
4)
1.
66
7(
2)
82
.1
6(
7)
87
.9
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
10
2
−1
09
.4
64
4(
4)
1.
14
9(
2)
85
.8
5(
7)
91
.9
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
+
N
2
,2
21
9
−1
09
.4
69
7(
4)
1.
08
8(
3)
86
.8
2(
7)
92
.9
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
+
N
2
,2
+
N
3
,0
26
0
−1
09
.4
70
2(
3)
1.
08
3(
2)
86
.9
1(
5)
93
.0
(1
)
N
2
,0
+
N
1
,1
+
A
z 1
,1
36
−1
09
.3
77
0(
6)
3.
67
0(
2)
69
.9
(1
)
74
.9
(2
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
A
z 1
,1
89
−1
09
.4
66
0(
3)
1.
11
6(
2)
86
.1
4(
5)
92
.2
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
A
z 1
,1
+
A
z
2
1
,1
97
−1
09
.4
66
9(
3)
1.
07
3(
2)
86
.3
1(
5)
92
.4
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
A
z 1
,1
+
A
z 2
,1
14
2
−1
09
.4
70
7(
3)
1.
07
2(
2)
87
.0
0(
5)
93
.1
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
A
z 1
,1
+
A
z
2
1
,1
+
A
z 2
,1
+
A
z
2
2
,1
19
1
−1
09
.4
71
4(
3)
1.
03
6(
4)
87
.1
3(
5)
93
.3
(1
)
V
M
C
(S
D
)
fr
om
R
ef
.
[1
35
]a
−1
09
.4
52
0(
5)
83
.5
9(
9)
89
.5
(2
)
D
M
C
−1
09
.5
06
0(
7)
93
.4
(1
)
10
0.
0(
2)
E
x
ac
t
fr
om
R
ef
.
[1
38
]
−1
09
.5
42
1
10
0
a
F
or
r N
N
=
2.
07
5
a.
u
.
W
e
d
o
n
ot
ex
p
ec
t
th
at
th
is
sm
al
l
d
iff
er
en
ce
in
b
on
d
le
n
gt
h
w
il
l
aff
ec
t
th
e
co
m
p
ar
is
on
b
et
w
ee
n
en
er
gi
es
si
gn
ifi
ca
n
tl
y.
75
4.4 RESULTS
(a) (b)0.2
0
−0.2
0.4
0
−0.4
figure 4.2: Plots of the (a) Az1,1 term and (b) N1,2 term for N2 as a function of the
position of an electron in a 12 a.u. × 12 a.u. plane containing the nuclei, indicated by
black circles.
This proportion is greater than the 93.0(1)% retrieved by our best isotropic Jastrow factor
J = N2,0 + N1,1 + N2,1 + N1,2 + N2,2 + N3,0, which includes more costly e–e–n–n and e–
e–e correlations and contains 260 optimizable parameters. We conclude that anisotropic
functions are an important tool in the construction of compact Jastrow factors for strongly
anisotropic systems.
Toulouse and Umrigar obtained 90% of the DMC correlation energy with a single-
determinant wave function [135], and with our best Jastrow factor we retrieve 93% of
the DMC correlation energy. We have also optimized a single-determinant backflow wave
function with our best Jastrow factor and we obtain a VMC energy of −109.4820(6) a.u.
(89% of the correlation energy), which is of similar accuracy to the multi-determinant
VMC energy of −109.4851(3) a.u. (89.6% of the correlation energy) obtained by Toulouse
and Umrigar.
table 4.11: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the N2 molecule.
N2,0 N1,1 N1,2 N2,1 N3,0 N2,2 A
s.h.
1,1 A
s.h.
2,1
p 9 – – 4 5 5 – 4
q – 9 7 4 – 3 9 4
4.4.3.3 H2O molecule
Single-particle spin-unrestricted HF orbitals for the 1A1 ground state of H2O were gen-
erated using the crystal Gaussian basis set code [141]. The basis set for O contains
14 s-, 9 p-, and 4 d-functions, and that for H contains 8 s-, 4 p-, and 3 d-functions.
Electron-nucleus cusps have been added using the scheme of Ma et al. [142]. We have
simulated a water molecule with a bond length of rOH = 1.8088 a.u. and a bond angle
76
4.4 RESULTS
of ∠HOH = 104.52◦ [143]. Our VMC results for different Jastrow factors are given in
Table 4.12 along with relevant reference energies.
Adding an N1,2 term to J = N2,0 + N1,1 + N2,1 gives only a very small improvement
for H2O, compared with the more substantial improvements obtained with this term for
BeH and N2. The N1,2 term acts as a correction to the single-electron orbitals, and we
believe that it is unimportant in this case because we have used very accurate HF orbitals,
whereas the single-electron orbitals used for BeH and N2 are considerably less accurate.
We find additional small improvements to the energy of H2O from adding N3,0 and N3,1
terms to J = N2,0 +N1,1 +N2,1.
Clark et al. obtained 92% of the DMC correlation energy with a single-determinant
wave function in Ref. [144], and with our best Jastrow factor we recover 95.5% of the
DMC correlation energy.
table 4.12: Energies (E) and VMC variances (V ) for the H2O molecule using different
Jastrow factors. We have used a bond length of rOH = 1.8088 a.u. and a bond angle of
∠HOH = 104.52◦ [143].
Np E (a.u.) V (a.u.) fCE (%) fDCE (%)
HF limit from Ref. [138] −76.0672 0 0
UHF (crystal) −76.0667 −0.1348 −0.1407
N2,0 18 −76.1640(6) 3.603(7) 26.1(2) 27.2(2)
N2,0+N1,1 36 −76.3368(3) 3.066(3) 72.71(8) 75.86(9)
N2,0+N1,1+N1,2 94 −76.3373(3) 3.051(6) 72.84(8) 76.00(9)
N2,0+N1,1+N2,1 266 −76.4030(2) 0.87(1) 90.56(5) 94.49(6)
N2,0+N1,1+N2,1+N1,2 325 −76.4035(2) 0.812(4) 90.70(5) 94.63(6)
N2,0+N1,1+N2,1+N1,2+N3,0 410 −76.4053(2) 0.829(5) 91.18(5) 95.13(6)
N2,0+N1,1+N2,1+N1,2+N3,1 741 −76.4068(2) 0.794(6) 91.59(5) 95.55(6)
VMC from Ref. [143] −76.3773(2) 83.63(5) 87.25(6)
VMC from Ref. [145] −76.3803(4) 84.4(1) 88.1(1)
VMC from Ref. [144] −76.3938(4) 88.1(1) 91.9(1)
DMC −76.4226(1) 95.85(3) 100.00(4)
Exact from Ref. [143] −76.438 100
table 4.13: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the H2O molecule.
N2,0 N1,1 N1,2 N2,1 N3,0 N3,1
p 9 – – 5 5 3
q – 9 7 5 – 3
4.4.3.4 H2 singlet
We studied the 1Σ+g singlet spin ground state at the equilibrium bond length rHH =
1.4011 a.u. [136] using cusp-corrected [142] HF orbitals. These orbitals were generated
77
4.4 RESULTS
by optimizing the coefficients and exponents of 13 s-, 6 p- and 4 d-functions using the
crystal Gaussian basis set code [141]. Our results are presented in Table 4.14.
10-3 10-2 10-1
VMC variance (a.u.)
10-3
10-2
E V
M
C
−
E 0
(a.
u.)
HF
N2,0
N2,0+N1,1
N2,0+N1,1+N1,2
N2,0+N1,1+N2,1
N2,0+N1,1+N2,1+N1,2
N2,0+N1,1+N2,1+N1,2+N2,2
figure 4.3: Difference between the VMC and exact energy against the VMC variance
for the H2 singlet spin ground state using different Jastrow factors.
The N2,0 term was able to recover a larger proportion of correlation energy in VMC for
the H2 singlet state than was possible for other systems. When the N1,1 term is included,
the wave function was able to recover over 95% of the correlation energy. This reflects
the simple electronic structure of the system.
Upon addition of the N2,1 term to J = N2,0 +N1,1, the N2,0 term e–e cut-off increases
from 6.3 a.u. to 7.3 a.u. while the N1,1 e–n cut-off increase from 5.1 a.u. to 5.8 a.u. The
N2,1 e–n cut-off optimizes to 4.0 a.u. These changes in cut-off values demonstrate that the
N2,1 term is necessary to differentiate length scales even in systems as homogeneous as H2.
The additional variational freedom provided by the N2,1 term yields a chemically-accurate
ground-state energy for the singlet state of H2.
The addition of the N1,2 term to either J = N2,0 +N1,1 or J = N2,0 +N1,1 +N2,1 does
not lead to a significant decrease in energy but the variance is reduced in the latter case.
We conclude that deficiencies in the basis set are largely ameliorated by the homogeneous
N1,1 term.
The Jastrow factor N2,0 +N1,1 +N2,1 +N1,2 +N2,2 contains all possible terms that can
78
4.4 RESULTS
t
a
b
l
e
4.
14
:
E
n
er
gi
es
(E
)
an
d
V
M
C
va
ri
an
ce
s
(V
)
fo
r
th
e
H
2
m
ol
ec
u
le
in
th
e
si
n
gl
et
sp
in
gr
ou
n
d
st
at
e
u
si
n
g
d
iff
er
en
t
J
as
tr
ow
fa
ct
or
s.
W
e
h
av
e
u
se
d
a
b
on
d
le
n
gt
h
of
r H
H
=
1.
40
11
a.
u
.
[1
36
]. N
p
E
(a
.u
.)
V
(a
.u
.)
f C
E
(%
)
f D
C
E
(%
)
H
F
li
m
it
fr
om
R
ef
.
[1
38
]
−1
.1
33
6
0
0
H
F
(c
r
y
st
a
l
)
−1
.1
33
6
0
0
N
2
,0
9
−1
.1
61
1(
2)
0.
02
25
3(
5)
67
.3
(5
)
67
.4
(5
)
N
2
,0
+
N
1
,1
18
−1
.1
72
69
(7
)
0.
00
66
0(
1)
95
.6
(2
)
95
.8
(2
)
N
2
,0
+
N
1
,1
+
N
1
,2
34
−1
.1
72
78
(8
)
0.
00
65
7(
1)
95
.9
(2
)
96
.0
(2
)
N
2
,0
+
N
1
,1
+
N
2
,1
76
−1
.1
74
01
(4
)
0.
00
17
05
(3
)
98
.9
(1
)
99
.0
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
92
−1
.1
74
06
(3
)
0.
00
14
91
(3
)
98
.9
8(
7)
99
.1
(1
)
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
+
N
2
,2
16
0
−1
.1
74
33
(2
)
0.
00
09
56
(2
)
99
.6
4(
5)
99
.8
(1
)
V
M
C
fr
om
R
ef
.
[1
28
]a
−1
.1
74
47
56
8(
21
)
0.
00
00
01
1
99
.9
99
4(
5)
10
0.
1(
1)
D
M
C
−1
.1
74
42
(4
)
99
.9
(1
)
10
0.
0(
1)
E
x
ac
t
fr
om
R
ef
.
[1
46
]
−1
.1
74
47
59
31
39
9(
1)
10
0.
0
a
F
or
r H
H
=
1.
4
a.
u
.
W
e
d
o
n
ot
ex
p
ec
t
th
at
th
is
sm
al
l
d
iff
er
en
ce
in
b
on
d
le
n
gt
h
w
il
l
aff
ec
t
th
e
co
m
p
ar
is
on
b
et
w
ee
n
en
er
gi
es
si
gn
ifi
ca
n
tl
y.
T
h
is
ca
lc
u
la
ti
on
cr
it
ic
al
ly
d
iff
er
s
fr
om
ou
rs
in
th
at
it
d
o
es
n
ot
en
fo
rc
e
th
e
cu
sp
co
n
d
it
io
n
s.
79
4.4 RESULTS
be constructed for this system and gives a VMC energy within error bars of the DMC
energy, which is exact for this system. We conclude that our terms are well-parametrized
and account for all the variational freedom needed for a Jastrow factor for the H2 singlet
state.
table 4.15: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the H2 singlet.
N2,0 N1,1 N1,2 N2,1 N2,2
p 9 – – 5 4
q – 9 6 5 3
4.4.3.5 H2 triplet
The energy of the first triplet spin excited state (3Σ+u ) of H2 has a very shallow minimum
corresponding to a large bond length of nearly 8 a.u. Although the exchange interaction
falls exponentially with increasing inter-nuclear separation, Kolos and Wolniewicz found
that it contributed significantly to the energy even at the large distance of 10 a.u. [147].
The strong interplay between the attractive dispersion forces and the repulsive exchange
interaction requires that both be accounted for to afford an accurate description of the
triplet state. This makes the system appealing for studying the construction of four-body
Jastrow factor terms to describe van der Waals-like interactions.
We used numerical HF orbitals tabulated on an elliptical grid obtained from the 2dhf
package [148] that were kindly generated by John Trail. HF theory predicts no binding
for the triplet state at any separation, and therefore any binding that occurs in VMC can
be attributed to the Jastrow factor. Unlike the singlet state, the nodal surface of this
state is not determined by symmetry and therefore DMC does not give the exact energy.
We have studied the H2 molecule in the triplet spin state at the inter-nuclear distance of
7.8358 a.u. This separation and the corresponding reference energy of −1.0000208957 a.u.
were found by fitting a quadratic function to the data of Staszewska and Wolniewicz [149].
In a preliminary study, we studied the molecule at a variety of other inter-nuclear dis-
tances in addition to the equilibrium distance, including the singlet spin state equilibrium
distance 1.401 a.u., 2.0 a.u., 4.0 a.u. and 6.0 a.u.
Equilibrium inter-nuclear distance
Previous QMC calculations on H2 at different inter-atomic distances have used Jastrow
factors with up to four-body correlations where the cusp conditions were not enforced
[127, 128], instead relying on the variance minimization method to find parameter values
that approximately satisfy the cusp conditions. This was found to be advantageous for
this system because the additional variational freedom yielded a better description in
80
4.4 RESULTS
VMC than when the cusp conditions were obeyed exactly [129]. The violation of the
cusp conditions is potentially catastrophic in DMC calculations, but these studies have
restricted the use of such terms to VMC.
We have optimized Jastrow factors consisting of the single e–e–n–n terms V2,2, F
b=1
2,2 ,
and B2,2 (see Table 4.1) at several expansion orders, where no constraints are enforced
at e–e or e–n coalescence points. We have used variance minimization for these Jastrow
factors as we found that it produces better results than energy minimization. The results
for the single-term Jastrow factors are given in Table 4.16. We have also optimized Jastrow
factors consisting of different sums of terms which satisfy the cusp conditions using energy
minimization. The results are given in Table 4.17 and are shown graphically in Fig. 4.4.
table 4.16: Energies (E) and VMC variances (V ) for H2 in the triplet spin state at
a bond length of rHH = 7.8358 a.u. using different cusp-violating single-term Jastrow
factors.
p q Np E (a.u.) V (a.u.) fCE (%) fDCE (%)
HF limit −0.9999828277 0 0
V2,2 0 4 11 −1.0000045(4) 0.0000205(1) 57(1) 57(1)
3 3 19 −1.0000100(3) 0.0000153(2) 71.4(8) 71.7(8)
4 3 25 −1.0000130(3) 0.000013(2) 79.3(8) 79.6(8)
0 7 29 −1.0000090(3) 0.0000152(1) 68.8(8) 69.1(8)
3 4 31 −1.0000139(3) 0.0000115(1) 81.6(8) 82.0(8)
4 4 41 −1.0000154(2) 0.0000083(1) 85.6(5) 86.0(6)
3 5 46 −1.0000157(2) 0.00000789(6) 86.4(5) 86.8(6)
4 5 61 −1.0000166(2) 0.0000066(1) 88.7(5) 89.2(6)
6 5 91 −1.0000175(2) 0.0000060(2) 91.1(5) 91.5(6)
5 6 106 −1.0000178(2) 0.000008(1) 91.9(5) 92.3(6)
B2,2 4 4 10 −1.0000086(3) 0.00001317(2) 67.7(8) 68.0(8)
5 5 21 −1.0000179(1) 0.00000347(1) 92.1(3) 92.6(4)
6 6 43 −1.00001966(8) 0.000001215(8) 96.8(2) 97.2(3)
7 7 79 −1.00002012(6) 0.00000067(1) 98.0(2) 98.4(3)
8 8 139 −1.00002028(5) 0.000000360(8) 98.4(1) 98.9(3)
9 9 229 −1.00002039(4) 0.000000268(7) 98.7(1) 99.2(3)
10 10 364 −1.00002045(3) 0.00000021(1) 98.83(8) 99.3(3)
F b=12,2 2 2 15 −1.0000176(2) 0.00000490(3) 91.3(5) 91.8(6)
3 3 82 −1.00001986(6) 0.000000718(7) 97.3(2) 97.8(3)
4 4 305 −1.00002037(3) 0.000000269(5) 98.62(8) 99.1(3)
DMC −1.0000207(1) 99.5(3) 100.0(4)
Exacta −1.0000208957 100
a Exact energy obtained by fitting to the data of Ref. [149].
We have performed the DMC calculations using our best B2,2 Jastrow factor and obtain
a reference DMC energy of −1.0000207(1) a.u. We have not encountered any statistical
problems in the DMC calculations with this cusp-violating wave function. Such issues can
occur when the local energy has a negative divergence in a region of configuration space
81
4.4 RESULTS
8 64 512
Np
10-7
10-6
10-5
E V
M
C
−
E 0
(a.
u.)
Multi-term
V2,2
F2,2
b=1
B2,2
DMC
figure 4.4: Difference between the VMC and exact energy against the number of wave
function parameters for the H2 triplet ground state using different Jastrow factors. Only
the multi-term Jastrow factor enforces the cusp conditions. The error bars are smaller
than the size of the symbol where not shown. All of the wave functions used here predict
binding.
82
4.4 RESULTS
table 4.17: Energies (E) and VMC variances (V ) for H2 in the triplet spin state at a
bond length of rHH = 7.8358 a.u. using different multi-term Jastrow factors.
Np E (a.u.) V (a.u.) fCE (%) fDCE (%)
HF limit −0.9999828277 0 0
N2,0 9 −0.9999994(4) 0.00001932(1) 44(1) 44(1)
N2,0+V2,2 25 −1.0000175(2) 0.000005434(7) 91.1(5) 91.5(6)
N2,0+N1,1 18 −1.0000106(3) 0.00001074(1) 73.0(8) 73.3(8)
N2,0+N1,1+V2,2 34 −1.0000180(2) 0.00000538(1) 92.4(5) 92.8(6)
N2,0+N1,1+N1,2 34 −1.0000133(2) 0.00000969(1) 80.0(5) 80.4(6)
N2,0+N1,1+N1,2+V2,2 50 −1.0000180(2) 0.000005250(7) 92.4(5) 92.8(6)
N2,0+N1,1+N2,1 45 −1.0000177(2) 0.00000476(1) 91.6(5) 92.1(6)
N2,0+N1,1+N2,1+V2,2 61 −1.0000192(1) 0.000003035(9) 95.5(3) 96.0(4)
N2,0+N1,1+N2,1+N1,2 61 −1.0000186(1) 0.00000351(1) 94.0(3) 94.4(4)
N2,0+N1,1+N2,1+N1,2+V2,2 77 −1.0000195(1) 0.000002108(6) 96.3(3) 96.8(4)
DMC −1.0000207(1) 99.5(3) 100.0(4)
Exacta −1.0000208957 100
a Exact energy obtained by fitting to the data of Ref. [149].
with a significant probability of being sampled. We have verified that our wave function
causes a negative divergence in the local energy when an electron coalesces with a nucleus,
leading us to conclude that the region of influence of this divergence is sufficiently small
that statistical problems do not arise in practice.
The F b=12,2 and B2,2 terms only differ in that the latter uses Boys-Handy-style indexing,
which yields slightly lower VMC energies than standard indexing in most cases for a fixed
number of parameters. Our best F b=12,2 and B2,2 Jastrow factors retrieve 99% of the DMC
correlation energy in VMC.
The V2,2 term is designed to describe van der Waals correlations, and contains e–e
functions which introduce other correlations. Our best V2,2 term recovers 92% of the
DMC correlation energy, offering a good description of the system without reaching the
accuracy of the more generic F b=12,2 and B2,2 terms.
A V2,2 term without e–e functions consists of contributions proportional to riI · rjJ ,
where the prefactors depend explicitly on riI and rjJ , and implicitly on rIJ . This func-
tional form is that of a dipole-dipole interactions. Our best such V2,2 term retrieves 69%
of the DMC correlation energy, which amounts to 0.0000262(3) a.u., and we regard this
as a measure of the pure van der Waals correlation energy of this system.
The multi-term Jastrow factors contain the usual N2,0, N1,1, N1,2, and N2,1 terms, and
for each combination of these we have added a V2,2 term without e–e functions obeying
the cusp conditions to study its effect. J = N2,0 retrieves 44% of the DMC correlation
energy, and adding the V2,2 term retrieves 85% of the remaining DMC correlation energy.
The effectiveness of V2,2 progressively drops as more terms are added, and it retrieves 43%
of the remaining DMC correlation energy when added to J = N2,0 + N1,1 + N2,1 + N1,2.
83
4.4 RESULTS
In all cases, V2,2 is found to lower the VMC energy by a larger amount than any of the
Nn,m terms.
Our best multi-term cusp-enforcing Jastrow factor retrieves 97% of the DMC correla-
tion energy with 77 wave-function parameters, comparable with the 98% retrieved with
the cusp-violating F b=12,2 and B2,2 terms with a similar number of parameters. For larger
systems where van der Waals interactions are important, we expect the violation of cusp
conditions to cause statistical problems, and the V2,2 term would become an effective way
of improving the description of the system in a multi-term Jastrow factor.
table 4.18: e–e and e–n expansion orders (p and q, respectively) used for the differ-
ent Jastrow factor terms in the multi-term Jastrow factors for the H2 triplet state at
7.8358 a.u.
N2,0 N1,1 N1,2 N2,1 V2,2
p 9 – – 4 0
q – 9 6 4 18
Various inter-nuclear distances
In our first study of the H2 triplet state, we optimized the Jastrow factors term-by-term
in two different sequences. In one set of optimizations, we started with the N2,0 term
and subsequently added the N1,1 term, three-body terms and finally the V2,2 term. In
the second set, we began with the V2,2 term and then proceeded to add the N2,0, N1,1
and three-body terms. We observe small differences in the energy obtained using the two
optimization sequences. Starting with the V2,2 term gives lower energies for d = 6 a.u.
and 7.836 a.u. while starting with N2,0 is preferable at shorter inter-nuclear distances. We
consider this to be reasonable as the N2,0 term becomes less important at larger distances
while van der Waals contributions described by V2,2 term become more important. The
percentage of the correlation energy recovered by each Jastrow factor for each distance is
given in Table 4.19. The reference energies for all distances are obtained from the data
of Ref. [149].
We first considered V2,2 terms without an expansion in e–e distances. This resulted in
poorer quality wave functions, and all further investigations included a small expansion in
e–e distances. As the addition of N2,2 to J = V2,2+N2,0+N1,1+N2,1+N1,2 at d = 7.836 a.u.
did not lead to further improvement, the N2,2 was not included in any other calculations.
A Slater determinant comprising Gaussian orbitals was also tested. However, the
variances of the energy were a factor of 25–100 times larger than those obtained with
numerical orbitals. Furthermore binding was not observed for any combination of terms.
This is likely due to the decay of the orbitals at large distance as e−r
2
instead of the
correct exponential decay. This leads to substantial noise in the large-distance regions
resulting in poor optimization of the Jastrow parameters.
84
4.4 RESULTS
table 4.19: Correlation energy retrieved (%) for the H2 triplet at various inter-nuclear
distances d (a.u.) using different Jastrow factors.
d = 1.401 2.0 4.0 6.0 7.836
N2,0 56.6(1) 53.9(1) 43.2(3) 40.8(6) 43(1)
N2,0+N1,1 85.86(7) 76.0(1) 66.3(2) 66.2(5) 75(1)
N2,0+N1,1+N1,2 90.55(5) 86.02(7) 76.4(2) 73.7(5) 80.6(8)
N2,0+N1,1+N2,1 91.95(5) 84.07(7) 86.6(1) 92.2(3) 94.2(5)
N2,0+N1,1+N2,1+N1,2 95.56(3) 92.28(5) 90.8(1) 94.2(2) 96.9(3)
N2,0+N1,1+N2,1+N1,2+V2,2 97.04(2) 95.25(5) 95.58(9) 97.1(2) 96.1(3)
V2,2 59.3(1) 59.1(1) 66.7(2) 71.7(5) 73(1)
V2,2+N2,0 81.85(7) 82.3(1) 90.7(1) 92.0(3) 91.6(5)
V2,2+N2,0+N1,1 91.01(5) 86.74(7) 90.7(1) 92.4(3) 91.3(5)
V2,2+N2,0+N1,1+N1,2 95.60(3) 92.33(7) 91.1(1) 93.8(2) 92.4(5)
V2,2+N2,0+N1,1+N2,1 95.53(3) 92.96(5) 94.5(1) 97.1(2) 96.1(3)
V2,2+N2,0+N1,1+N2,1+N1,2 97.04(2) 95.20(5) 95.0(1) 97.9(1) 97.1(3)
DMC 98.56(3) 99.29(5) 99.3(1) 99.8(2) 99.9(4)
The cusp conditions are not satisfied by a Jastrow factor consisting of only the V2,2
term. The addition of the N2,0 term, which satisfies the cusp conditions, reduces the
variance by a factor of 3–5 for all inter-atomic distances.
The N1,2 term is found to give a non-negligible improvement at all distances, particu-
larly at intermediate distances for J = N2,0+N1,1+N1,2. In the presence of the V2,2 term,
both the N1,1 and N1,2 terms have a greater impact at d = 1.401 a.u. and 2 a.u. than at
larger distances. The effect of the V2,2 term on the charge density is small enough that the
N1,1 and N1,2 basis-set correction terms together retrieve only 0.4–14% more correlation
energy. On the other hand, these terms allow J = N2,0+N1,1+N1,2 to recover 33–37%
more correlation energy than J = N2,0.
We recover over 95% of the correlation energy at VMC level for all distances. Datta
et al. [150] are able to recover 99.951(2)% of the correlation energy at VMC level for a
separation of d = 2 a.u. using a highly-accurate exponential Hylleraas-type form. Our
equilibrium distance DMC energy is well within an error bar of the exact energy. It
appears that the quality of the HF nodal surface improves with increasing inter-nuclear
distance.
For a small system such as the H2 molecule, visualizing the Jastrow factor gives insight
into its evolution as higher-order terms are added. We have plotted the contribution of an
electron to the Jastrow function as it is scanned across the plane of the molecule for various
Jastrow factors. The nuclei, which we label A and B are separated by the equilibrium
bond length of 7.836 a.u. and the second electron is fixed at a distance 1.958 a.u. from
nucleus A perpendicular to the bond. The Jastrow factor augments the Slater determinant
contribution to the wave function in the blue regions and diminishes it in the red regions.
In all cases, the Jastrow functions decay to zero as the electron is moved far from the
85
4.4 RESULTS
molecule and the correct asymptotic behaviour of the one-electron orbitals is retained.
In Fig. 4.5, we plot the Jastrow factor as each term is sequentially added, giving
the final J = N2,0+N1,1+N2,1+N1,2+V2,2 Jastrow factor. The sequential construction of
J = V2,2+N2,0+N1,1+N2,1+N1,2 is depicted in Fig. 4.6.
≤-0.3
-0.2
-0.1
0
0.1(a) (b) (c)
(d) (e) (f)
figure 4.5: Plots of the (a) N2,0, (b) N2,0+N1,1, (c) N2,0+N1,1+N2,1, (d) N2,0+N1,1+N1,2,
(e) N2,0+N1,1+N2,1+N1,2, (f) N2,0+N1,1+N2,1+N1,2+V2,2 Jastrow functions. The nuclei
are indicated by black circles and the fixed electron is indicated by a cross.
The isotropic N2,0 term reduces the probability of finding the two electrons close
together, as can be seen in Figs. 4.5(a) and 4.6(b). The addition of the N1,1 term pulls
the charge density away from the region of the fixed electron into the region near nucleus
B in Fig. 4.5(b). By increasing the value of the wave function on the opposite side of
nucleus A from the fixed electron, the N2,1 term makes the system more ionic. This
behaviour is observed for inter-nuclear distances of 4 a.u. and 6 a.u. as well as for J =
V2,2+N2,0+N1,1+N2,1+N1,2, as seen in Fig. 4.6(d). The V2,2 term does not contribute
significantly when optimized last (Fig. 4.5(f)), as discussed below.
We also studied the contributions of each term to the final Jastrow factor for each
optimization sequence. Plots of the term-wise contributions are given in Fig. 4.7. We see
that while the magnitude of the contribution of the N2,0, N1,1, N2,1 and N1,2 terms varies,
their qualitative shapes remain the same. However, the V2,2 term (Fig. 4.7(e)) varies enor-
mously when optimized first compared to when optimized last, resulting in qualitatively
very different total Jastrow factors (Fig. 4.7(f)). The V2,2+N2,0+N1,1+N2,1+N1,2 Jastrow
factor is dominated by the V2,2 term while the N2,0+N1,1+N2,1+N1,2+V2,2 Jastrow factor
is unaffected by its presence. The difference in magnitude and structure of the V2,2 term
86
4.4 RESULTS
≤-0.3
-0.2
-0.1
0
0.1
(a) (b) (c)
(d) (e) (f)
figure 4.6: Plots of the (a) V2,2, (b) V2,2+N2,0, (c) V2,2+N2,0+N1,1,
(d) V2,2+N2,0+N1,1+N2,1, (e) V2,2+N2,0+N1,1+N1,2, (f) V2,2+N2,0+N1,1+N2,1+N1,2 Jas-
trow functions. The nuclei are indicated by black circles and the fixed electron is indicated
by a cross.
is highlighted in Fig. 4.8. We believe that the importance of the sequence in which terms
are optimized is a result of the well-known difficulty that VMC energy optimization has
in optimizing cut-off lengths4. During term-by-term optimization of Jastrow factors, we
recommend optimizing important terms first (i.e., those that recover a larger fraction of
the correlation energy), and subsequently adding less important terms.
table 4.20: e–e and e–n expansion orders (p and q, respectively) used for the different
Jastrow factor terms in the multi-term Jastrow factors for the H2 molecule at various
distances.
N2,0 N1,1 N1,2 N2,1 V2,2
p 9 – – 5 3
q – 9 6 5 18
4.4.4 Discussion of molecular results
In Fig. 4.9 we have plotted the fraction of the DMC correlation energy retrieved by
different Jastrow factor terms for BeH, N2, H2O, and the H2 singlet and triplet states.
4Shortly after this study was complete, a bug in the energy minimization routine resulting in a less
than optimal minimization was found and fixed. It is possible that this dependence on optimization
sequence is now weaker or even non-existent.
87
4.4 RESULTS
(a
)
-0
.6
-0
.30
(b
)
-0
.0
6
-0
.0
3
00.
03
(c
)
0 0.
1
0.
2
(d
)
00.
04
0.
08
(e
)
-0
.0
6
-0
.0
3
00.
03
(f
)
-0
.1
-0
.0
5
00.
05
f
ig
u
r
e
4.
7:
P
lo
ts
of
th
e
(a
)
N
2
,0
(b
)
N
1
,1
(c
)
N
2
,1
(d
)
N
1
,2
(e
)
N
2
,2
te
rm
s,
an
d
(f
)
th
e
to
ta
l
J
as
tr
ow
fu
n
ct
io
n
.
T
h
e
fi
rs
t
co
lu
m
n
co
rr
es
p
on
d
s
to
an
op
ti
m
iz
at
io
n
se
q
u
en
ce
of
V
2
,2
+
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
an
d
th
e
se
co
n
d
co
lu
m
n
co
rr
es
p
on
d
s
to
an
op
ti
m
iz
at
io
n
se
q
u
en
ce
of
N
2
,0
+
N
1
,1
+
N
2
,1
+
N
1
,2
+
V
2
,2
.
T
h
e
n
u
cl
ei
ar
e
in
d
ic
at
ed
b
y
b
la
ck
ci
rc
le
s
an
d
th
e
fi
x
ed
el
ec
tr
on
is
in
d
ic
at
ed
b
y
a
cr
os
s.
88
4.4 RESULTS
(a) (b)
0.03
0
−0.03
−0.06
2× 10−4
0
−2× 10−4
figure 4.8: Plots of the V2,2 term when optimized in the (a) V2,2+N2,0+N1,1+N2,1+N1,2
sequence and the (b) N2,0+N1,1+N2,1+N1,2+V2,2 sequence. The nuclei are indicated by
black circles and the fixed electron is indicated by a cross.
Our purpose is to visualize the importance of different terms in different systems, and to
this end we do not include anisotropic or cusp-violating terms.
The N2,0 term represents the simplest description of electronic correlations and typi-
cally retrieves 20–25% of the DMC correlation energy. This e–e term distorts the charge
density of the HF wave function, and the N1,1 term repairs this, typically retrieving
an additional 45–50% of the DMC correlation energy. In the case of the more diffuse H2
molecule the N2,0 and N1,1 terms have a different relative importance. The J = N2,0 +N1,1
factor recovers about 95% of the DMC correlation energy for the H2 singlet and 70–75%
of the DMC correlation energy in the other four molecules.
Like N1,1, N1,2 acts as a correction to the single-electron orbitals. This term provides
no significant benefit in H2O, where we have used high-quality orbitals, but it recovers
7% of the DMC correlation energy for the H2 triplet. A visual comparison of the N1,1 and
N1,2 terms for the H2 triplet is given in Fig. 4.10. It is clear that the N1,2 term is largely
acting in the bond region of the molecule, where there is overlap of the isotropic N1,1
terms centred at the two nuclei. Introduction of the N1,2 term allows N1,1 to be better
optimized further away from the bond.
Clearly, the behaviour of the N1,1 correction in the bond direction needs to be distin-
guished. This has been done in two ways in this work. Firstly, introducing a N1,2 term
recovers 4% more correlation energy for N2, making it the most important isotropic term
beyond the N2,0, N1,1 and N2,1 terms. The N1,2 term recovers between 4.5–10% of the H2
triplet correlation energy at various distances. Secondly, anisotropic cut-offs were used
to build in an explicit angular dependence to differentiate the bond direction in the N2
molecule. Using up to l = 2 spherical harmonics, an additional 5.4% of the correlation
energy was recovered.
The effect of N1,2 in N2 is noteworthy in that the energy reduction obtained by adding
this term to J = N2,0 + N1,1 is about a factor of four times smaller than when added to
89
4.4 RESULTS
BeH N2 H2O H2(S) H2(T)
0
20
40
60
80
100
f DC
E
(%
)
V2,2
N2,2
N3,1
N2,1
N1,2
N1,1
N2,0
figure 4.9: Fraction of the DMC correlation energy retrieved by different Jastrow factor
terms for the BeH, N2, H2O, H2 singlet and H2 triplet molecules at their equilibrium
geometries.
(a) (b)0.02
0
−0.02
0.06
0.03
0
figure 4.10: Plots of the (a) N1,1 and (b) N1,2 terms. The nuclei are indicated by black
circles.
90
4.5 CONCLUSIONS
the more accurate J = N2,0 + N1,1 + N2,1. One would expect a term to retrieve more
correlation energy when added to a smaller Jastrow factor, and this is the case for N1,2
in the other molecules. We think that the distortion in the charge density caused by N2,1
in N2 is such that the single-electron correction effected by N1,2 becomes more useful in
its presence.
The N2,1 term added to J = N2,0 + N1,1 + N1,2 captures an additional 15–20% of the
DMC correlation energy for BeH, H2O and N2. This demonstrates the importance of the
N2,1 term in systems with different length scales. The variation of the importance of the
N2,1 term with distance is made clear by the H2 triplet. At the short bond length of
1.401 a.u., the 21 term recovers about 6% of the correlation energy and this progressively
increases to about 26% at a bond length of 6 a.u.
Higher-order terms added to J = N2,0 + N1,1 + N2,1 + N1,2 yield significant gains
in relative terms, with e–e–n–n terms retrieving 13% and 43% of the remaining DMC
correlation energy remaining for N2 and the H2 triplet, respectively, and the e–e–e–n
term recovering 17% of the remaining DMC correlation energy for H2O.
4.4.5 Summary of results
Table 4.21 gives a comparison of the best single-determinant non-backflow VMC energies
we have found in the literature with those obtained in this work.
4.5 Conclusions
We have described a generalized Jastrow factor allowing terms that explicitly correlate
the motions of n electrons with m static nuclei. These terms can be parametrized using
various basis sets, including terms that involve dot products of inter-particle position
vectors. We have also introduced anisotropic cut-off functions. The formalism may be
applied to systems with particle types and external potentials other than electrons and
Coulomb potentials.
Optimization of the wave function is one of the most human- and computer-time
consuming tasks in performing QMC calculations. We have performed term-by-term
optimizations to understand how different terms in the Jastrow factor contribute to the
electronic description of a system, and we hope that our analysis will serve as a guideline
for constructing Jastrow factors for other systems.
We have tested these terms on HEGs, atoms, and molecules. The variational freedom
from the higher-order terms generally improves the quality of the wave function. It has
been argued that higher-order terms can be neglected [131, 126] as the Pauli exclusion
principle does not allow for more than two electrons to be close. Huang et al. [126]
suggest that it would be more economical to improve the wave function by including a
91
4.5 CONCLUSIONS
t
a
b
l
e
4.
21
:
B
es
t
si
n
gl
e-
d
et
er
m
in
an
t
n
on
-b
ac
k
fl
ow
V
M
C
en
er
gi
es
(a
.u
.)
fo
u
n
d
in
th
e
li
te
ra
tu
re
an
d
th
os
e
fr
om
th
is
w
or
k
,
al
on
g
w
it
h
si
n
gl
e-
d
et
er
m
in
an
t
D
M
C
an
d
ex
ac
t
en
er
gi
es
fo
r
re
fe
re
n
ce
.
S
y
st
em
T
h
is
w
or
k
L
it
er
at
u
re
D
M
C
E
x
ac
t
1D
H
E
G
(r
s
=
5
a.
u
.,
N
=
19
)
−0
.2
04
08
33
(2
)
−0
.2
04
08
34
(3
)
−0
.2
04
08
34
(3
)
2D
H
E
G
(r
s
=
35
a.
u
.,
N
=
42
)
−0
.0
27
61
12
(6
)
−0
.0
27
70
87
(1
)
B
e
−1
4.
65
22
(1
)
−1
4.
64
97
2(
5)
a
−1
4.
65
71
7(
4)
−1
4.
66
73
6
B
−2
4.
63
09
(2
)
−2
4.
62
93
6(
5)
a
−2
4.
64
00
2(
6)
−2
4.
65
39
1
O
−7
5.
03
81
(3
)
−7
5.
03
52
(1
)a
−7
5.
05
11
(1
)
−7
5.
06
73
B
eH
−1
5.
24
12
(3
)
−1
5.
22
8(
1)
b
−1
5.
24
60
3(
4)
−1
5.
24
82
N
2
−1
09
.4
71
4(
3)
−1
09
.4
52
0(
5)
c
−1
09
.5
06
0(
7)
−1
09
.5
42
1
H
2
O
−7
6.
40
68
(2
)
−7
6.
39
38
(4
)d
−7
6.
42
26
(1
)
−7
6.
43
8
H
2
si
n
gl
et
(1
Σ
+ g
)
−1
.1
74
33
(2
)
−1
.1
74
47
56
8(
21
)e
−1
.1
74
42
(4
)
−1
.1
74
47
59
31
39
9(
1)
H
2
tr
ip
le
t
(3
Σ
+ u
)
−1
.0
00
02
04
5(
3)
−1
.0
00
02
07
(1
)
−1
.0
00
02
08
95
7
a
R
ef
.
[1
35
].
b
R
ef
.
[1
40
].
c
R
ef
.
[1
35
]
(u
si
n
g
a
sl
ig
h
tl
y
d
iff
er
en
t
b
on
d
le
n
gt
h
).
d
R
ef
.
[1
44
].
e
R
ef
.
[1
28
]
(u
si
n
g
a
sl
ig
h
tl
y
d
iff
er
en
t
b
on
d
le
n
gt
h
).
92
4.5 CONCLUSIONS
multi-determinant wave function than by using higher-order Jastrow terms, specifically a
e–e–e–n term. This is often correct but we have concentrated on using a single determinant
as our primary goal was to study the Jastrow factor. Of course, our Jastrow factor can
be used with other wave function forms. It would be interesting to study whether the
hypothesis of Huang et al. extends to terms other than the N3,1 term.
We have demonstrated the construction and application of an e–e–n–n Jastrow factor
term designed to describe van der Waals interactions between atoms. This term retrieves
a large fraction of the van der Waals correlation energy in tests on the triplet state of H2
at the proton separation of minimum total energy.
We have found evidence for the importance of three-electron Jastrow terms in the low-
density 1D and 2D HEGs. Improving the Jastrow factor for single-determinant backflow
wave functions also leads to improvements in the DMC energy of the 2D HEG. This
demonstrates the indirect effect that improving the Jastrow factor can have on improving
the nodal surface, as reported in Ref. [124].
We have made efforts to obtain accurate single-determinant VMC energies for most of
the systems studied, but for BeH and N2 we deliberately used inferior one-electron basis
sets to see whether we could compensate for this with one-electron Jastrow terms. We
find that this goal can be achieved by including an N1,2 Jastrow term or anisotropic e–n
terms, along with the usual N1,1 term.
In strongly inhomogeneous systems, the N1,2 term is shown to be important in describ-
ing the bond region, allowing the N1,1 term to correct the basis set far from the bond.
It is conceivable that more compact representations can be constructed by considering
bond-centred terms. This idea is motivated by the bond-centred orbitals developed by
the quantum chemistry community, and would be an interesting basis for future work.
93
Chapter 5
Orbital-Dependent Backflow
Transformations
5.1 Introduction
F
undamentally, the DMC method is limited by the fixed-node approximation that
must be made to overcome the fermion sign problem. Consequently, the DMC en-
ergy is limited by the accuracy of the nodal surface of the trial wave function. The
Jastrow factor is everywhere positive and cannot modify the nodes. Improving nodes
therefore relies on improving the orbital component of the wave function, such as af-
forded by a multi-determinant expansion or pairing wave functions. Evaluating orbitals
at backflow-transformed quasiparticle coordinates can also achieve this. These backflow
transformations are the topic of this chapter.
Backflow transformations can be motivated as improvements to the one-electron or-
bitals used in a wave function consisting of a single Slater determinant. One-electron
orbitals do not allow for a description of correlation, but such a wave function is able to
describe exchange exactly as a result of the built-in antisymmetry. To account for the
anti-parallel spin correlation hole, Wigner and Seitz [151] used a wave function where the
up-spin electron orbitals depended parametrically on the positions of the down-spin elec-
trons. The form of the wave function used by Wigner and Seitz is related to the backflow
wave function introduced by Feynman [152] and Feynman and Cohen [153]. Feynman and
Cohen extended classical backflow, which is related the flow of an incompressible fluid
around an impurity, to excitations in pure liquid helium and the 4He system with 3He
impurities. For these systems, backflow achieves a flow pattern that conserves the local
current and increases the effective impurity mass.
The Slater wave function written as a product of up- and down-spin determinants
lacks direct spin coupling as the probability of finding the up-spin electrons in a given
configuration is independent of the position of the down-spin electrons and vice versa.
The form of backflow transformations used in QMC calculations remedies this deficiency.
94
5.1 INTRODUCTION
The orbitals comprising the Slater determinant
ΨD(X(R)) =
∣∣∣∣∣∣∣∣∣∣
ψ1(x1) ψ1(x2) · · · ψ1(xN)
ψ2(x1) ψ2(x2) · · · ψ2(xN)
...
...
. . .
...
ψN(x1) ψN(x2) · · · ψN(xN)
∣∣∣∣∣∣∣∣∣∣
(5.1)
are evaluated at the backflow-transformed quasiparticle coordinates xi which are a func-
tion of the positions of all electrons,
xi = ri + ξi(R). (5.2)
The contribution to the displacement from anti-parallel spin electrons is found to be
larger [69]. Vitiello et al. [154] compared the effect of backflow transformations to that of
a spin-dependent Jastrow factor. They found that both give similar results, demonstrating
a possible equivalence between backflow and spin-dependent correlations.
For homogeneous systems such as electron gases, the backflow function ξi(R) is taken
to be a function of inter-electron separations. The presence of nuclei introduces inho-
mogeneity into the system which to some extent is included via electron-nucleus terms.
Higher-order electron-electron-nucleus terms are also found to be particularly important
for inhomogeneous systems [69]. The inhomogeneous backflow function developed by
Lo´pez R´ıos et al. [69] is
ξi(R) =
N∑
j 6=i
η(rij)rij +
M∑
I
µ(riI)riI
+
N∑
j 6=i
M∑
I
[
ΦI(rij, riI , rjI)rij + Θ
I(rij, riI , rjI)riI
]
,
(5.3)
where η is the e–e term, µ is the e–n term and ΦI and ΘI are e–e–n terms.
Backflow transformations are useful in DMC calculations because they can improve the
nodal surface. However, this improvement comes at a price. As the backflow-transformed
position of each electron is a function of the position of all the other electrons, changing
the coordinate of one electron changes the transformed coordinates of all electrons. Each
orbital must then be evaluated for each electron configuration. This significantly increases
the cost of QMC calculations which then scale asO(N4) rather than asO(N3). In practice,
the added cost of including backflow transformations is lower since only particles within
a cut-off distance contribute.
Backflow has also been argued to represent momentum-dependent correlation [155].
Hence it is natural to consider backflow transformations specific to orbitals representing
different momentum states. This motivates the development of new backflow transfo-
95
5.2 IMPLEMENTATION
mations presented here, which are able to accommodate different parametrizations of
the backflow function for distinct orbitals. The Slater determinant with these orbital-
dependent backflow transformations is then
ΨD(X(R)) =
∣∣∣∣∣∣∣∣∣∣
ψt1(x
t
1) ψ
t
1(x
t
2) · · · ψt1(xtN)
ψu2 (x
u
1) ψ
u
2 (x
u
2) · · · ψu2 (xuN)
...
...
. . .
...
ψvN(x
v
1) ψ
v
N(x
v
2) · · · ψvN(xvN)
∣∣∣∣∣∣∣∣∣∣
, (5.4)
where the indices t, u and v represent backflow parameter set indices and
xni = ri + ξ
n
i (R). (5.5)
For t = u = . . . = v = 1, we recover orbital-independent backflow. This work is still
in progress and the benefits of these orbital-dependent backflow transformations in im-
proving the wave function are yet to be assessed. In the following sections, the required
modifications to the algorithms and the rise of additional variational freedom is discussed.
5.2 Implementation
The main changes that need to be made to the existing structure of casino to sup-
port orbital-dependent backflow transformations relate to the way in which orbitals are
indexed, evaluated and used. These changes in orbital management also affect the evalu-
ation of the kinetic energy.
5.2.1 Management of orbitals
Firstly, a list of unique orbitals is constructed. An orbital map is used to map rows
of different determinants to the appropriate orbital index. Rather than updating one
entire column of a Slater matrix whenever an electron is moved, the orbital index and
map structure allow sections of a column to be updated. This is necessary for orbital-
dependent backflow as discussed below.
The orbitals must then be classified into groups that have the same backflow trans-
formations. Generally, they can be classified by a number of quantities. For example,
plane-wave orbitals can be characterized by their k-vector and band, or by their eigen-
value. Atomic orbitals can be labelled by their principal quantum number n, angular
momentum quantum number l and magnetic quantum number m. For each type of basis
set, we have constructed a list of quantities that can be used to characterize the orbitals.
Each unique orbital is then labelled by its characteristics. The user specifies which of these
characteristics is to be used to distinguish orbitals for the purpose of orbital-dependent
96
5.2 IMPLEMENTATION
backflow transformations. Orbitals sharing a characteristic value form a group and all
such orbitals are assigned the same transformation index n, allowing the application of
different backflow transformations ξni (R) to different groups of orbitals.
The orbital map and transformation index data are used to construct an orbital mask,
which indicates whether a given orbital belongs to the given transformation group1. How-
ever, using an orbital mask requires looping over all orbitals to identify those belonging
to a group. This operation is computationally inefficient in the cases where only a few
orbitals belong to a given group. It can be more efficient to construct orbital ranges for
each transformation that store the first and last indices of a sequence of orbitals belonging
to the same group.
The construction of the Slater matrix is modified by the existence of orbital-dependent
quasiparticle coordinates. An element of the Slater matrix is denoted by ψκ,nlj = ψ
κ,n
l (x
n
j ),
where ψκ,nl represents a one-electron orbital with transformation index n in the l
th row of
the κth determinant and the quasiparticle coordinate is evaluated using the nth backflow
transformation. Instead of evaluating all orbitals at each of the n sets of quasiparticle
coordinates, it is more efficient to loop over the transformation index n and evaluate all the
corresponding orbitals ψn with transformation index n, regardless of the determinants in
which they appear, at the appropriate backflow-transformed electron coordinates xn. The
orbital map is then used to update the Slater matrices appropriately for all determinants
with the new orbital values.
These modifications are useful not only in the implementation of orbital-dependent
backflow transformations but also for non-backflow calculations. They serve to simplify
evaluation routines significantly and unify the underlying structure of wave function eval-
uation. A speed-up has been observed in the evaluation of certain wave function types,
e.g., by avoiding repeated evaluation of orbitals for different determinants. Additionally,
they lay the foundations for the integration of other more complex types of wave functions
such as geminals and pfaffians into the casino code.
5.2.2 Kinetic energy evaluation
The total kinetic energy of a system is the sum of the kinetic energy of all the electrons,
K =
N∑
i
Ki =
N∑
i
−1
2
Ψ−1∇2iΨ. (5.6)
For several reasons, the kinetic energy of each electron is evaluated as
Ki = 2Ti − |Fi|2, (5.7)
1Here we are only interested in grouping orbitals based on their transformation index. Orbitals are
also grouped by spin, and the grouping can easily be extended to other characteristics.
97
5.2 IMPLEMENTATION
where
Ti = −1
4
∇2i ln |Ψ| = −
1
4
[
∇2iΨD
ΨD
−
(∇iΨD
ΨD
)2
+∇2iJ
]
(5.8)
and
Fi = − 1√
2
∇i ln |Ψ| = − 1√
2
(∇iΨD
ΨD
+∇iJ
)
. (5.9)
Firstly, 〈Ki〉 = 〈|Fi|2〉 = 〈Ti〉 in VMC. Violation of this condition is indicative of prob-
lems in a VMC calculation such as a bug in the code. Secondly, the contribution of the
determinantal part of the wave function ΨD is separated from the contribution of the
Jastrow factor, allowing modularization of the code. In what follows, we are only inter-
ested in calculating the derivatives of ΨD as the Jastrow factor is unaffected by backflow
transformations.
The basic quantities required for calculating the contribution of the determinantal
part of wave function to the local energy are Mαi = ∇αi ln |ΨD| and Ni = ∇2i ln |ΨD| which
respectively appear in Ti and Fi. In the derivations below, Greek letters α, β and γ
represent Cartesian component indices, n and m are transformation indices, l and q are
orbital indices and i, j and p are electron indices.
The determinantal component of a multi-determinant-backflow wave function2 is
ΨD =
∑
k
ck
S∏
σ
Dk,σ, (5.10)
where the kth determinant is written as a product of determinants Dk,σ of sets of distin-
guishable particles with index σ. Then,
Mαi = ∇αi ln |ΨD| =
1
ΨD
∑
k
ck
∏
σ
Dk,σ
∑
τ
∇αi Dk,τ
Dk,τ
(5.11)
and
Ni = ∇2i ln |ΨD| = −
(∇iΨD
ΨD
)2
+
∇2iΨD
ΨD
= −|Mi|2 + ∇
2
iΨD
ΨD
(5.12)
where
∇2iΨD =
∑
k
ck
∏
σ
Dk,σ
∑
α
(∑
τ
∇αi Dk,τ
Dk,τ
)2
−
∑
τ
(∇iDk,τ
Dk,τ
)2
+
∑
τ
∇2iDk,τ
Dk,τ
. (5.13)
Once again, care must be taken in the evaluation of ∇αi Dκ and ∇2iDκ as the quasipar-
ticle positions at which the orbitals are evaluated depend on the backflow transformation
index of the orbital. Note that the determinant index k and particle group index σ have
2The corresponding expressions for other types of wave functions such as geminals and pfaffians will
be different.
98
5.3 VARIATIONAL FREEDOM
been absorbed into a single index κ below. This index explicitly indicates the dependence
of the Slater matrix elements on the determinant. Rather than summing over all orbitals
l, we sum over all transformations and restrict the sum over orbitals to those that belong
to the given transformation group, i.e.,
∑
l →
∑
m
∑
l∈m.
In this notation, ∇αi Dκ and ∇2iDκ are given by
∇αi Dκ =
∂Dκ
∂rαi
=
∑
m
l∈m
j
∂Dκ
∂ψκ,mlj
∑
β
∂ψκ,mlj
∂xm,βj
∂xm,βj
∂rαi
(5.14)
and
∇2iDκ =
∑
α
∂
∂rαi
∑m
l∈m
j
∂Dκ
∂ψκ,mlj
∑
β
∂ψκ,mlj
∂xm,βj
∂xm,βj
∂rαi
=
∑
m
l∈m
j
∂Dκ
∂ψκ,mlj
∑
β
∂ψκ,mlj
∂xm,βj
∑
α
∂2xm,βj
∂(rαi )
2
+
∑
m
l∈m
j
∂Dκ
∂ψκ,mlj
∑
β,γ
∂2ψκ,mlj
∂xm,βj ∂x
m,γ
j
∑
α
∂xm,βj
∂rαi
∂xm,γj
∂rαi
+
∑
m
l∈m
j
∑
n
q∈m
p
∂2Dκ
∂ψκ,mlj ∂ψ
κ,n
qp
∑
β,γ
∂ψκ,mlj
∂xm,βj
∂ψκ,nqp
∂xn,γj
∂xm,βj
∂rαi
∂xn,γj
∂rαi
, (5.15)
where
∂2Dκ
∂ψκ,mlj ∂ψ
κ,n
qp
=
1
Dκ
[
∂Dκ
∂ψκ,mlj
∂Dκ
∂ψκ,nqp
− ∂Dκ
∂ψκ,mlp
∂Dκ
∂ψκ,nqj
]
. (5.16)
5.3 Variational freedom
In addition to the variational freedom introduced by allowing different backflow parameter
sets for different orbitals, we also have further freedom in the choice of orbitals in the
Slater matrix when using orbital-dependent backflow transformations. This freedom arises
because the Slater determinant is no longer invariant under linear transformation of the
ψ orbital basis.
This freedom in the choice of linear combinations of orbitals is not present for wave
functions using the traditional orbital-independent backflow transformations. Consider
99
5.3 VARIATIONAL FREEDOM
the Slater determinant
D0 =
∣∣∣∣∣∣∣∣∣∣
ψ1(x1) ψ1(x2) · · · ψ1(xN)
ψ2(x1) ψ2(x2) · · · ψ2(xN)
...
...
. . .
...
ψN(x1) ψN(x2) · · · ψN(xN)
∣∣∣∣∣∣∣∣∣∣
. (5.17)
We can combine ψ1 with some proportion c2 of ψ2 to give a new determinant
D =
∣∣∣∣∣∣∣∣∣∣
ψ1(x1) + c2ψ2(x1) ψ1(x2) + c2ψ2(x2) · · · ψ1(xN) + c2ψ2(xN)
ψ2(x1) ψ2(x2) · · · ψ2(xN)
...
...
. . .
...
ψN(x1) ψN(x2) · · · ψN(xN)
∣∣∣∣∣∣∣∣∣∣
(5.18)
=
∣∣∣∣∣∣∣∣∣∣
[ψ1 + c2ψ2](x1) [ψ1 + c2ψ2](x2) · · · [ψ1 + c2ψ2](xN)
ψ2(x1) ψ2(x2) · · · ψ2(xN)
...
...
. . .
...
ψN(x1) ψN(x2) · · · ψN(xN)
∣∣∣∣∣∣∣∣∣∣
. (5.19)
It is easy to show that D = D0 using the properties of determinants.
The determinant is not, however, necessarily invariant when linear combinations of
orbitals are used with orbital-dependent backflow. Consider
D0 =
∣∣∣∣∣∣∣∣∣∣
ψt1(x
t
1) ψ
t
1(x
t
2) · · · ψt1(xtN)
ψu2 (x
u
1) ψ
u
2 (x
u
2) · · · ψu2 (xuN)
...
...
. . .
...
ψvN(x
v
1) ψ
v
N(x
v
2) · · · ψvN(xvN)
∣∣∣∣∣∣∣∣∣∣
. (5.20)
Again, we can combine ψ1 with some proportion c2 of ψ2 to construct a new determinant
D =
∣∣∣∣∣∣∣∣∣∣
ψt1(x
t
1) + c2ψ
u
2 (x
u
1) ψ
t
1(x
t
2) + c2ψ
u
2 (x
u
2) · · · ψt1(xtN) + c2ψu2 (xuN)
ψu2 (x
u
1) ψ
u
2 (x
u
2) · · · ψu2 (xuN)
...
...
. . .
...
ψvN(x
v
1) ψ
v
N(x
v
2) · · · ψvN(xvN)
∣∣∣∣∣∣∣∣∣∣
. (5.21)
Unless t = u, the linear combination ψt1(x
t
i)+c2ψ
u
2 (x
u
i ) is a function of two quasiparticle co-
ordinates. We have the freedom to construct a new determinant D′ of single-quasiparticle
100
5.4 SYSTEMS OF INTEREST
orbitals by assigning the orbital ψt1 + c2ψ
u
2 a new transformation index s:
D′ =
∣∣∣∣∣∣∣∣∣∣
[ψt1 + c2ψ
u
2 ]
s(xs1) [ψ
t
1 + c2ψ
u
2 ]
s(xs2) · · · [ψt1 + c2ψu2 ]s(xsN)
ψu2 (x
u
1) ψ
u
2 (x
u
2) · · · ψu2 (xuN)
...
...
. . .
...
ψvN(x
v
1) ψ
v
N(x
v
2) · · · ψvN(xvN)
∣∣∣∣∣∣∣∣∣∣
. (5.22)
We can thus optimize the orbitals that comprise the Slater matrix in addition to the
backflow functions for each of these orbitals. Generalizing to an arbitrary combination of
orbitals, we can write
D =
∣∣∣∣∣∣∣∣∣∣
1 c12 · · · c1N
c21 1 · · · c2N
...
...
. . .
...
cN1 cN2 · · · 1
ψt1(x
t
1) ψ
t
1(x
t
2) · · · ψt1(xtN)
ψu2 (x
u
1) ψ
u
2 (x
u
2) · · · ψu2 (xuN)
...
...
. . .
...
ψvN(x
v
1) ψ
v
N(x
v
2) · · · ψvN(xvN)
∣∣∣∣∣∣∣∣∣∣
(5.23)
where cij determines the amplitude of ψi in the j
th transformed orbital. Each of the cij
can be optimized from its initial value of 0 subject to the constraint that the resulting
orbitals are linearly independent. This is equivalent to demanding that the matrix of
coefficients be non-singular: ∣∣∣∣∣∣∣∣∣∣
1 c12 · · · c1N
c21 1 · · · c2N
...
...
. . .
...
cN1 cN2 · · · 1
∣∣∣∣∣∣∣∣∣∣
6= 0. (5.24)
This condition is checked during optimization and parameter sets that do not satisfy it
are rejected.
5.4 Systems of interest
These modifications are expected to be valuable in studying systems with one-electrons
orbitals of very different characters. The 3D HEG is one such system. At high densities,
backflow effects are known to become more important [156]. The uniform zero-energy
k = 0 state differs significantly from the oscillating high-energy states. Given this variation
in orbital character, we expect the optimal backflow transformations for each orbital to
vary as well.
Systems such as TiO2 might also benefit from the use of orbital-dependent transfor-
mations. The localized d-orbital character of the valence Ti electrons is very different
from the more diffuse character of the valence p-electron orbitals in O.
101
5.5 SUMMARY
5.5 Summary
Orbital-dependent backflow transformations are expected to be more suitable for obtain-
ing an accurate trial wave function than the system-averaged transformations currently
used. Primarily, they will further improve the nodal surface of the wave function and
thus bring DMC energies closer to the exact energies. It would be interesting to see if a
less complex parametrization of the backflow terms, namely a smaller polynomial expan-
sion, would suffice when using orbital-dependent backflow transformations. This would
help limit the cost of including these transformations and allow for better optimization.
While the benefits of using other basis functions and higher-order terms are expected to
be small, they will be investigated in further work.
102
Chapter 6
Conclusions
T
he focus of this thesis is the use of improved wave functions to perform highly-
accurate QMC calculations of finite and extended systems.
QMC calculations of the first-row atoms Li–Ne and their singly-positively-charged
ions are reported. Multi-determinant-Jastrow-backflow trial wave functions recovered
more than 98% of the correlation energy at the VMC level and more than 99% of the
correlation energy at the DMC level for both the atoms and ions. We obtained the first
ionization potentials to chemical accuracy for all atoms. Scalar relativistic corrections to
the energies, mass-polarization terms, and one- and two-electron expectation values are
reported. Fits to the electron and intracule densities are also performed.
A flexible framework for constructing Jastrow factors which allows for the introduc-
tion of terms involving arbitrary numbers of particles is described. Jastrow factors in-
cluding various three- and four-body terms, a four-body van der Waals-like term, and
anisotropic terms are constructed. They are used in QMC calculations of the one- and
two-dimensional homogeneous electron gases, the Be, B, and O atoms, the BeH, H2O
and N2 molecules, and the singlet and triplet states of the H2 molecule. Our optimized
Jastrow factors retrieve more than 90% of the DMC correlation energy in VMC for each
system studied.
Orbital-dependent backflow transformations are motivated. Their implementation in
the casino QMC code is described. We expect orbital-dependent backflow transforma-
tions to play an important role in improving the nodal surface in systems with large
variations in orbital character. Finally, some systems are suggested as candidates for
testing these transformations.
103
References
[1] M. Born and R. Oppenheimer, Ann. Phys. 87, 457 (1927).
[2] A. Szabo´ and N. S. Ostlund, Modern quantum chemistry: introduction to advanced
electronic structure theory (Dover Publications, New York, 1996).
[3] T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic-structure theory (Wi-
ley, Chichester, 2000).
[4] G. H. Booth, A. J. W. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009).
[5] D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 132, 041103 (2010).
[6] G. H. Booth and A. Alavi, J. Chem. Phys. 132, 174104 (2010).
[7] D. M. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 134, 024112 (2011).
[8] J. J. Shepherd, G. Booth, A. Gru¨neis, and A. Alavi, Phys. Rev. B 85, 081103
(2012).
[9] J. S. Spencer, N. S. Blunt, and W. M. C. Foulkes, J. Chem. Phys. 136, 054110
(2012).
[10] C. Ha¨ttig, W. Klopper, A. Ko¨hn, and D. P. Tew, Chem. Rev. 112, 4 (2011).
[11] S. F. Boys and N. C. Handy, Proc. R. Soc. London, Ser. A 309, 209 (1969).
[12] S. F. Boys and N. C. Handy, Proc. R. Soc. London, Ser. A 310, 43 (1969).
[13] S. F. Boys and N. C. Handy, Proc. R. Soc. London, Ser. A 310, 63 (1969).
[14] S. F. Boys and N. C. Handy, Proc. R. Soc. London, Ser. A 311, 309 (1969).
[15] S. Tsuneyuki, Prog. Theor. Phys. Suppl. 176, 134 (2008).
[16] S. Ten-no, Chem. Phys. Lett. 330, 169 (2000).
[17] N. Umezawa and S. Tsuneyuki, J. Chem. Phys. 119, 10015 (2003).
104
REFERENCES
[18] N. Umezawa, S. Tsuneyuki, T. Ohno, K. Shiraishi, and T. Chikyow, J. Chem. Phys.
122, 224101 (2005).
[19] R. G. Parr and W. Yang, Density-functional theory of atoms and molecules (Oxford
University Press, Oxford, 1994).
[20] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
[21] M. Levy, Proc. Natl. Acad. Sci. 76, 6062 (1979).
[22] E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983).
[23] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
[24] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
[25] D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).
[26] G. Senatore and N. H. March, Rev. Mod. Phys. 66, 445 (1994).
[27] S. Baroni and S. Moroni, Phys. Rev. Lett. 82, 4745 (1999).
[28] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73,
33 (2001).
[29] M. P. Nightingale and C. J. Umrigar, Quantum Monte Carlo methods in physics
and chemistry (Kluwer Academic Publishers, Dordrecht, 1999).
[30] B. Hammond, W. Lester, and P. Reynolds, Monte Carlo methods in ab initio
quantum chemistry (World Scientific, Singapore, 1994).
[31] R. J. Needs, M. D. Towler, N. D. Drummond, and P. Lo´pez R´ıos, J. Phys.: Condens.
Matter 22, 023201 (2010).
[32] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical
recipes: the art of scientific computing (Cambridge University Press, Cambridge,
2007).
[33] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J.
Chem. Phys. 21, 1087 (1953).
[34] D. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 16, 3081 (1977).
[35] P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, J. Chem. Phys. 77,
5593 (1982).
[36] R. P. Feynman, Statistical mechanics: a set of lectures (Addison Wesley Longman,
Reading, 1998).
105
REFERENCES
[37] M. Troyer and U. Wiese, Phys. Rev. Lett. 94, 170201 (2005).
[38] J. B. Anderson, J. Chem. Phys. 63, 1499 (1975).
[39] J. B. Anderson, J. Chem. Phys. 65, 4121 (1976).
[40] R. Grimm and R. Storer, J. Comput. Phys. 7, 134 (1971).
[41] M. H. Kalos, D. Levesque, and L. Verlet, Phys. Rev. A 9, 2178 (1974).
[42] D. M. Ceperley, J. Stat. Phys. 63, 1237 (1991).
[43] C. J. Umrigar, M. P. Nightingale, and K. J. Runge, J. Chem. Phys. 99, 2865 (1993).
[44] D. Ceperley, M. H. Kalos, and J. L. Lebowitz, Macromolecules 14, 1472 (1981).
[45] N. Nemec, Phys. Rev. B 81, 035119 (2010).
[46] R. Barnett, P. Reynolds, and W. Lester Jr, J. Computat. Phys. 96, 258 (1991).
[47] J. R. Trail and R. Maezono, J. Chem. Phys. 133, 174120 (2010).
[48] G. Ortiz, D. M. Ceperley, and R. M. Martin, Phys. Rev. Lett. 71, 2777 (1993).
[49] N. D. Drummond, PhD thesis, University of Cambridge, Cambridge, 2004.
[50] R. Jastrow, Phys. Rev. 98, 1479 (1955).
[51] G. Weerasinghe, personal communication.
[52] P. Lo´pez R´ıos, personal communication.
[53] A. C. Hurley, J. Lennard-Jones, and J. A. Pople, Proc. R. Soc. London, Ser. A 220,
446 (1953).
[54] M. Casula and S. Sorella, J. Chem. Phys. 119, 6500 (2003).
[55] M. Marchi, S. Azadi, M. Casula, and S. Sorella, J. Chem. Phys. 131, 154116 (2009).
[56] M. Bajdich, L. Mitas, G. Drobny´, L. K. Wagner, and K. E. Schmidt, Phys. Rev.
Lett. 96, 130201 (2006).
[57] M. Bajdich, L. Mitas, L. K. Wagner, and K. E. Schmidt, Phys. Rev. B 77, 115112
(2008).
[58] A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. E 71, 066704
(2005).
[59] L. Mitas, E. L. Shirley, and D. M. Ceperley, J. Chem. Phys. 95, 3467 (1991).
106
REFERENCES
[60] M. Casula, Phys. Rev. B 74, 161102 (2006).
[61] M. Casula, S. Moroni, S. Sorella, and C. Filippi, J. Chem. Phys. 132, 154113 (2010).
[62] S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holzmann, Phys. Rev. Lett. 97,
076404 (2006).
[63] N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C. Foulkes, Phys. Rev. B
78, 125106 (2008).
[64] H. Flyvbjerg and H. G. Petersen, J. Chem. Phys. 91, 461 (1989).
[65] J. R. Trail, Phys. Rev. E 77, 016703 (2008).
[66] J. R. Trail, Phys. Rev. E 77, 016704 (2008).
[67] C. J. Umrigar, Phys. Rev. Lett. 71, 408 (1993).
[68] R. M. Lee, G. J. Conduit, N. Nemec, P. Lo´pez R´ıos, and N. D. Drummond, Phys.
Rev. E 83, 066706 (2011).
[69] P. Lo´pez R´ıos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs, Phys.
Rev. E 74, 066701 (2006).
[70] M. Dewing, J. Chem. Phys. 113, 5123 (2000).
[71] N. D. Drummond, Z. Radnai, J. R. Trail, M. D. Towler, and R. J. Needs, Phys.
Rev. B 69, 085116 (2004).
[72] N. D. Drummond and R. J. Needs, Phys. Rev. Lett. 102, 126402 (2009).
[73] C. J. Umrigar, K. G. Wilson, and J. W. Wilkins, Phys. Rev. Lett. 60, 1719 (1988).
[74] P. R. C. Kent, R. J. Needs, and G. Rajagopal, Phys. Rev. B 59, 12344 (1999).
[75] N. D. Drummond and R. J. Needs, Phys. Rev. B 72, 085124 (2005).
[76] M. Snajdr and S. M. Rothstein, J. Chem. Phys. 112, 4935 (2000).
[77] F. J. Ga´lvez, E. Buend´ıa, and A. Sarsa, J. Chem. Phys. 115, 1166 (2001).
[78] A. Badinski and R. J. Needs, Phys. Rev. E 76, 036707 (2007).
[79] D. Ceperley, J. Stat. Phys. 43, 815 (1986).
[80] M. P. Nightingale and V. Melik-Alaverdian, Phys. Rev. Lett. 87, 043401 (2001).
[81] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev.
Lett. 98, 110201 (2007).
107
REFERENCES
[82] J. Toulouse and C. J. Umrigar, J. Chem. Phys. 126, 084102 (2007).
[83] C. Froese Fischer, G. Tachiev, G. Gaigalas, and M. R. Godefroid, Comput. Phys.
Commun. 176, 559 (2007).
[84] N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. B 70, 235119 (2004).
[85] M. D. Brown, J. R. Trail, P. Lo´pez R´ıos, and R. J. Needs, J. Chem. Phys. 126,
224110 (2007).
[86] N. D. Drummond, P. Lo´pez R´ıos, A. Ma, J. R. Trail, G. G. Spink, M. D. Towler,
and R. J. Needs, J. Chem. Phys. 124, 224104 (2006).
[87] M. Puchalski and K. Pachucki, Phys. Rev. A 78, 052511 (2008).
[88] S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. A. Parpia, and C. Froese
Fischer, Phys. Rev. A 47, 3649 (1993).
[89] J. R. Trail, personal communication.
[90] E. Buend´ıa, F. J. Ga´lvez, P. Maldonado, and A. Sarsa, J. Chem. Phys. 131, 044115
(2009).
[91] K. Hongo, Y. Kawazoe, and H. Yasuhara, Mater. Trans. 47, 2612 (2006).
[92] R. Prasad, N. Umezawa, D. Domin, R. Salomon-Ferrer, and W. A. Lester, J. Chem.
Phys. 126, 164109 (2007).
[93] W. Klopper, R. A. Bachorz, D. P. Tew, and C. Ha¨ttig, Phys. Rev. A 81, 022503
(2010).
[94] F. De Proft and P. Geerlings, J. Chem. Phys. 106, 3270 (1997).
[95] M. Ernzerhof and G. E. Scuseria, J. Chem. Phys. 110, 5029 (1999).
[96] Q. Zhao and R. G. Parr, J. Chem. Phys. 98, 543 (1993).
[97] Q. Zhao, R. C. Morrison, and R. G. Parr, Phys. Rev. A 50, 2138 (1994).
[98] E. Steiner, J. Chem. Phys. 39, 2365 (1963).
[99] M. Levy, J. P. Perdew, and V. Sahni, Phys. Rev. A 30, 2745 (1984).
[100] R. Assaraf, M. Caffarel, and A. Scemama, Phys. Rev. E 75, 035701 (2007).
[101] J. Katriel, Phys. Rev. A 5, 1990 (1972).
[102] R. A. Bonham and M. Fink, High-energy electron scattering (Van Nostrand Rein-
hold, New York, 1974).
108
REFERENCES
[103] R. J. Weiss, X-ray determination of electron distributions (Elsevier, Amsterdam,
1966).
[104] R. J. Boyd, C. Sarasola, and J. M. Ugalde, J. Phys. B 21, 2555 (1988).
[105] A. Sarsa, F. J. Ga´lvez, and E. Buend´ıa, J. Chem. Phys. 109, 7075 (1998).
[106] J. Toulouse, R. Assaraf, and C. J. Umrigar, J. Chem. Phys. 126, 244112 (2007).
[107] A. J. Thakkar and V. H. Smith Jr., Chem. Phys. Lett. 42, 476 (1976).
[108] A. J. Thakkar, J. Chem. Phys. 84, 6830 (1986).
[109] Y. Lee and R. J. Needs, unpublished.
[110] R. Assaraf and M. Caffarel, Phys. Rev. Lett. 83, 4682 (1999).
[111] S. D. Kenny, G. Rajagopal, and R. J. Needs, Phys. Rev. A 51, 1898 (1995).
[112] S. A. Alexander, S. Datta, and R. L. Coldwell, Phys. Rev. A 81, 032519 (2010).
[113] F. W. King, D. G. Ballegeer, D. J. Larson, P. J. Pelzl, S. A. Scott, T. J. Prosa, and
B. M. Hinaus, Phys. Rev. A 58, 3597 (1998).
[114] Z. Yan and G. W. F. Drake, Phys. Rev. Lett. 81, 774 (1998).
[115] C. L. Pekeris, Phys. Rev. 126, 1470 (1962).
[116] K. Pachucki and J. Komasa, Phys. Rev. Lett. 92, 213001 (2004).
[117] T. Koga and H. Matsuyama, J. Chem. Phys. 115, 3984 (2001).
[118] F. W. King, J. Chem. Phys. 102, 8053 (1995).
[119] Z. Yan and G. W. F. Drake, Phys. Rev. A 52, 3711 (1995).
[120] A. M. Frolov, J. Chem. Phys. 124, 224323 (2006).
[121] A. M. Frolov and D. M. Wardlaw, J. Exp. Theor. Phys. 108, 583 (2009).
[122] F. J. Ga´lvez and I. Porras, Phys. Rev. A 44, 144 (1991).
[123] A. J. Cohen, N. C. Handy, and B. O. Roos, Phys. Chem. Chem. Phys. 6, 2928
(2004).
[124] Y. Kwon, D. M. Ceperley, and R. M. Martin, Phys. Rev. B 48, 12037 (1993).
[125] T. Kato, Commun. Pure Appl. Math. 10, 151 (1957).
109
REFERENCES
[126] C.-J. Huang, C. J. Umrigar, and M. P. Nightingale, J. Chem. Phys. 107, 3007
(1997).
[127] S. A. Alexander and R. L. Coldwell, J. Chem. Phys. 121, 11557 (2004).
[128] M. C. Per, S. P. Russo, and I. K. Snook, J. Chem. Phys. 130, 134103 (2009).
[129] M. C. Per, personal communication.
[130] K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, 4172 (1990).
[131] C. Filippi and C. J. Umrigar, J. Chem. Phys. 105, 213 (1996).
[132] P. Seth, P. Lo´pez R´ıos, and R. J. Needs, J. Chem. Phys. 134, 084105 (2011).
[133] N. D. Drummond and R. J. Needs, Phys. Rev. Lett. 99, 166401 (2007).
[134] R. M. Lee and N. D. Drummond, Phys. Rev. B 83, 245114 (2011).
[135] J. Toulouse and C. J. Umrigar, J. Chem. Phys. 128, 174101 (2008).
[136] D. Feller, K. A. Peterson, and D. A. Dixon, J. Chem. Phys. 129, 204105 (2008).
[137] G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. Fonseca Guerra, S. J. A. van Gis-
bergen, J. G. Snijders, and T. Ziegler, J. Comput. Chem. 22, 931 (2001).
[138] D. P. ONeill and P. M. W. Gill, Mol. Phys. 103, 763 (2005).
[139] A. Lu¨chow and J. B. Anderson, J. Chem. Phys. 105, 7573 (1996).
[140] N. Nemec, M. D. Towler, and R. J. Needs, J. Chem. Phys. 132, 034111 (2010).
[141] R. Dovesi, R. Orlando, B. Civalleri, C. Roetti, V. R. Saunders, and C. M. Zicovich-
Wilson Z. Kristallogr. 220, 571 (2005).
[142] A. Ma, M. D. Towler, N. D. Drummond, and R. J. Needs, J. Chem. Phys. 122,
224322 (2005).
[143] I. G. Gurtubay and R. J. Needs, J. Chem. Phys. 127, 124306 (2007).
[144] B. K. Clark, M. A. Morales, J. McMinis, J. Kim, and G. E. Scuseria, J. Chem.
Phys. 135, 244105 (2011).
[145] M. Casula, C. Attaccalite, and S. Sorella, J. Chem. Phys. 121, 7110 (2004).
[146] J. S. Sims and S. A. Hagstrom, J. Chem. Phys. 124, 094101 (2006).
[147] W. Koos and L. Wolniewicz, J. Chem. Phys. 43, 2429 (1965).
[148] L. Laaksonen, P. Pyykko, and D. Sundholm, Comput. Phys. Rep. 4, 313 (1986).
110
REFERENCES
[149] G. Staszewska and L. Wolniewicz, J. Mol. Spectrosc. 198, 416 (1999).
[150] S. Datta, S. A. Alexander, and R. L. Coldwell, Int. J. Quantum Chem. 111, 4106
(2011).
[151] E. Wigner and F. Seitz, Phys. Rev. 46, 509 (1934).
[152] R. P. Feynman, Phys. Rev. 94, 262 (1954).
[153] R. P. Feynman and M. Cohen, Phys. Rev. 102, 1189 (1956).
[154] S. A. Vitiello, K. E. Schmidt, and S. Fantoni, Phys. Rev. B 55, 5647 (1997).
[155] V. R. Pandharipande and N. Itoh, Phys. Rev. A 8, 2564 (1973).
[156] Y. Kwon, D. M. Ceperley, and R. M. Martin, Phys. Rev. B 58, 6800 (1998).
111
Supplementary Information
Electron- and intracule-density fitting
parameters
The parameters for the least-linear square fits to the binned electron and intracule den-
sities as described in Sec. 3.4.3 are given here. The number of parameters in each case
was chosen to minimize χ2 while giving a sensible density gradient as r → 0. The errors
in the normalization constants for the charge and intracule density fits are of O(10−3) or
smaller, except for the intracule densities for B and C, where they are of O(10−2).
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4
6
5
6
5
b 5
1
.5
1
9
3
0
7
0
0
.8
2
8
7
7
3
0
4
1
.4
5
1
7
0
5
3
2
.0
9
4
2
2
9
7
3
.3
1
1
2
6
3
2
3
.3
6
6
0
0
4
5
5
.1
5
9
7
0
9
1
5
.3
5
0
9
0
3
8
b 6
1
.5
1
9
1
3
6
0
0
.5
0
0
1
2
3
1
6
1
.0
5
1
4
6
8
2
1
.4
0
8
6
3
3
7
1
.9
3
5
9
0
4
5
3
.0
9
5
3
3
9
5
4
.1
4
1
1
7
4
3
−
b 7
−1
.0
8
9
4
5
7
8
−1
.0
5
4
2
9
7
2
1
.6
9
8
6
5
3
7
1
.2
0
8
6
4
9
9
0
.8
7
0
3
7
0
2
7
1
.7
1
8
1
3
6
6
2
.5
7
0
8
4
3
3
−
b 8
−1
.5
0
6
3
7
3
0
−0
.8
2
2
6
6
8
1
7
1
.0
3
9
3
7
9
2
1
.4
2
6
5
8
8
7
0
.9
6
9
0
9
9
2
7
0
.6
0
5
3
5
0
3
6
1
.3
7
8
6
8
2
3
−
b 9
0
.7
4
2
8
4
2
6
5
0
.7
9
5
6
8
1
4
3
0
.7
5
6
5
1
2
9
6
1
.0
2
8
1
1
2
1
0
.8
1
8
3
5
7
6
5
1
.0
2
0
3
4
9
4
1
.1
7
1
1
2
6
2
−
b 1
0
0
.0
9
5
7
7
1
4
7
0
0
.5
1
8
6
5
9
3
6
−
0
.8
1
5
6
2
5
4
8
−
0
.8
4
3
0
9
5
3
6
1
.1
4
4
8
1
1
7
−
b 1
1
1
.5
2
2
8
3
1
3
×
1
0
−
5
0
.7
2
2
0
7
3
6
6
−
0
.9
5
7
8
5
2
2
9
−
−
1
.0
1
7
5
1
2
8
−
b 1
2
0
.6
8
2
1
7
5
3
7
0
.6
9
2
4
7
3
3
3
−
0
.8
2
3
7
1
5
6
1
−
−
0
.7
3
8
4
8
3
1
1
−
b 1
3
−
0
.8
8
2
9
6
4
5
8
−
−
−
−
−
−
t
a
b
l
e
2:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
d
ow
n
-s
p
in
el
ec
tr
on
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
at
om
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
4.
L
i
B
e
B
C
N
O
F
N
e
A
0
.0
0
2
2
5
9
1
3
9
1
1
.0
7
2
4
5
8
0
×
1
0
−
6
−0
.0
7
0
6
0
7
8
4
4
0
.0
4
3
6
3
8
3
1
8
−0
.1
2
8
9
7
7
9
2
0
.0
2
2
3
2
7
3
6
5
−0
.0
0
4
6
3
8
6
7
9
6
−7
.5
1
6
8
0
0
8
×
1
0
−
8
a
0
−1
.9
1
6
6
8
8
3
−2
.8
7
1
7
1
1
2
−3
.5
6
9
8
4
7
6
−4
.1
3
2
5
1
2
3
−4
.4
8
5
7
1
0
0
−5
.0
2
5
1
9
4
5
−5
.3
9
9
0
3
7
2
−5
.7
4
4
0
2
7
1
a
2
−6
.1
4
3
5
1
4
6
−3
.1
7
9
2
9
6
8
−1
.4
9
0
2
3
4
1
−0
.6
0
9
1
7
5
9
8
−2
.0
5
0
2
5
9
8
−1
.4
0
0
4
2
4
6
−1
.9
8
0
6
4
7
1
1
.8
2
3
7
5
4
9
a
3
1
1
.8
3
6
9
3
3
2
.0
0
0
2
7
2
1
0
.5
3
7
0
4
2
9
0
−2
.4
8
9
4
6
9
9
0
.2
7
8
9
0
2
1
2
−8
.4
1
4
1
7
3
4
−2
4
.4
7
7
8
8
7
−7
5
.9
4
4
4
0
9
a
4
1
2
.2
5
8
3
1
1
7
.8
3
7
8
3
2
9
2
.2
7
3
1
5
8
3
−1
.4
9
3
1
5
2
2
−2
.5
9
6
4
0
5
2
−1
0
.5
0
8
2
2
4
−2
5
.6
6
5
8
1
2
−1
6
.2
9
5
8
3
6
a
5
4
.0
7
8
5
7
9
6
8
.0
0
4
9
8
0
8
1
.9
3
0
0
7
1
1
7
.9
4
7
5
6
2
4
1
1
.4
8
2
1
3
5
2
6
.0
5
9
2
7
5
5
4
.1
5
4
4
0
8
5
1
.6
7
9
1
0
6
a
6
−0
.4
0
3
0
4
6
2
9
4
.0
0
5
9
8
3
6
4
.6
5
3
1
3
4
4
1
5
.3
6
3
3
0
6
2
9
.0
7
3
9
0
9
5
5
.8
4
6
5
2
9
8
5
.4
9
0
5
3
4
1
2
9
.4
0
5
2
1
a
7
2
.4
6
1
9
6
8
9
−1
.1
0
4
1
5
5
8
5
.1
0
8
6
5
2
7
1
3
.2
7
6
6
5
4
2
2
.3
5
3
4
4
8
4
3
.0
5
2
7
4
5
4
9
.5
5
3
8
3
6
1
1
.7
1
2
8
1
6
a
8
0
.2
8
6
9
7
4
3
4
−1
.5
2
0
7
8
7
1
2
.3
7
6
7
7
7
1
8
.6
9
6
4
6
0
1
9
.2
9
6
7
4
4
3
1
8
.9
5
1
7
0
9
2
2
.4
7
0
9
5
6
−
a
9
−
3
.8
6
7
9
8
4
2
1
.7
6
2
3
8
7
4
7
.3
7
1
0
8
4
7
5
.8
0
5
8
8
8
6
8
.7
2
1
3
0
5
0
1
4
.3
9
1
7
9
9
−
a
1
0
−
6
.6
0
4
3
4
4
3
0
.2
4
9
9
1
1
1
3
6
.0
2
8
2
8
2
3
4
.1
5
1
3
1
3
2
6
.0
0
8
7
0
7
3
7
.0
4
0
4
8
1
0
−
a
1
1
−
2
.5
8
4
3
1
1
3
−
2
.9
9
3
6
1
0
5
1
.8
7
1
4
7
6
8
2
.6
9
1
4
0
3
9
1
.3
0
6
0
3
3
7
−
a
1
2
−
0
.7
2
1
1
1
5
5
5
−
1
.3
7
4
1
1
3
0
2
.1
1
4
8
6
1
0
1
.4
3
5
3
0
1
0
0
.8
1
2
6
4
5
0
9
−
a
1
3
−
2
.5
2
3
3
2
1
8
−
1
.7
4
9
0
0
2
0
0
.3
0
9
6
8
4
9
2
2
.2
9
0
0
8
3
9
1
.8
6
3
8
7
4
4
−
a
1
4
−
1
.1
8
8
5
5
0
0
−
0
.2
7
6
4
3
3
0
2
−
2
.0
6
0
2
8
6
8
0
.8
0
1
0
2
5
4
4
−
a
1
5
−
0
.5
8
5
7
9
4
1
3
−
−
−
0
.9
2
1
4
0
7
0
5
−
−
b 2
−1
.6
7
1
1
5
6
7
0
.8
6
3
8
4
1
4
0
0
.6
2
2
8
2
7
1
5
−0
.0
0
0
1
9
2
6
2
3
7
3
−0
.0
0
2
2
9
9
5
0
8
2
−0
.0
0
2
1
5
0
2
0
6
0
−0
.0
1
8
3
9
2
7
6
3
−0
.0
0
9
4
7
7
9
9
5
0
b 3
1
.7
5
1
8
0
3
1
1
.2
9
0
9
3
9
1
0
.1
5
3
7
4
9
7
4
0
.3
5
6
4
6
3
8
6
0
.0
5
3
9
2
8
1
7
0
−0
.0
0
1
2
9
4
8
2
2
8
−0
.0
0
1
9
8
0
6
0
5
0
−0
.1
5
0
7
7
2
9
4
b 4
0
.4
6
1
2
2
2
6
0
1
.1
3
2
5
1
7
5
0
.0
0
0
9
7
9
7
2
9
4
2
0
.8
7
0
0
8
3
5
5
1
.2
9
5
8
8
3
0
3
.3
4
4
7
5
7
8
6
.7
1
8
2
7
3
8
9
.4
4
3
4
2
4
7
b 5
0
.8
5
3
0
0
8
8
8
0
.2
5
8
0
5
0
1
1
1
.2
6
9
0
0
9
5
1
.7
3
2
6
7
9
9
3
.3
8
8
6
2
7
0
5
.0
6
9
9
7
5
8
5
.0
4
7
6
8
8
7
4
.3
6
6
6
2
8
3
b 6
0
.4
5
6
1
6
7
1
0
−0
.5
1
5
8
6
8
6
1
1
.1
9
7
5
1
3
7
2
.1
4
9
1
3
4
6
2
.7
5
4
6
8
3
5
4
.3
8
9
2
0
7
7
4
.4
2
1
4
1
1
6
−
b 7
−
−1
.0
7
8
4
5
4
2
1
.1
8
8
2
2
3
1
2
.2
4
9
1
5
4
2
−2
.1
9
9
2
3
7
6
2
.9
1
1
6
3
2
9
3
.8
7
7
8
2
3
8
−
b 8
−
0
.8
3
9
4
5
5
9
1
0
.8
8
1
3
2
2
2
3
1
.9
0
5
7
2
8
4
1
.4
3
1
2
1
6
8
1
.8
1
2
1
3
6
2
2
.2
9
0
2
6
1
4
−
b 9
−
0
.7
9
0
2
0
1
3
2
−
1
.1
7
5
9
5
0
9
0
.8
2
2
6
1
5
1
7
0
.2
8
4
8
9
2
0
1
0
.9
3
9
2
6
3
7
7
−
b 1
0
−
0
.6
4
4
0
1
4
9
6
−
0
.9
8
8
3
0
4
3
4
1
.1
0
4
7
2
6
2
1
.4
3
5
6
8
4
2
1
.0
3
8
7
5
8
2
−
b 1
1
−
0
.8
4
8
3
9
7
2
3
−
0
.9
4
0
0
2
9
0
6
0
.8
3
1
6
6
2
4
8
1
.2
5
8
6
7
0
1
0
.9
3
9
4
4
7
2
5
−
b 1
2
−
0
.6
6
5
6
1
6
6
6
−
0
.8
0
7
8
3
3
0
7
−
0
.9
0
3
3
7
8
0
4
0
.6
9
1
9
6
3
9
8
−
b 1
3
−
0
.8
8
5
6
0
4
1
2
−
−
−
0
.7
1
8
8
3
0
9
3
−
−
t
a
b
l
e
3:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
u
p
-s
p
in
el
ec
tr
on
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
io
n
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
4.
L
i+
B
e+
B
+
C
+
N
+
O
+
F
+
N
e+
A
0
.0
0
5
1
3
2
8
1
1
4
0
.3
8
3
0
0
8
4
2
−1
.3
9
5
7
0
1
9
×
1
0
−
6
0
.0
4
9
3
9
5
2
3
0
−7
.7
9
1
1
3
5
6
×
1
0
−
6
1
.6
8
7
1
9
2
2
×
1
0
−
6
0
.1
5
2
7
6
4
3
8
−1
.4
6
0
2
1
7
6
×
1
0
−
6
a
0
−1
.9
1
1
6
6
9
6
−2
.8
9
0
3
1
6
4
−3
.5
8
9
2
4
6
6
−4
.1
5
9
4
7
9
7
−4
.6
4
2
2
8
0
0
−5
.0
5
1
9
2
9
4
−5
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9
0
4
8
0
0
−5
.7
3
6
9
2
4
5
a
2
−1
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7
0
9
4
3
9
−0
.8
7
8
7
3
1
8
6
−1
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3
1
3
8
0
7
−0
.6
6
2
9
6
4
6
3
−1
.4
0
8
3
4
4
0
−1
.6
6
3
0
6
4
5
2
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0
2
1
4
3
3
−2
.0
5
1
0
9
6
3
a
3
1
.3
8
6
4
2
0
0
0
.5
8
3
7
3
6
9
3
0
.3
1
0
1
0
4
5
0
−4
.6
6
9
1
8
9
4
−1
2
.3
1
5
7
9
1
−2
6
.1
2
4
4
3
0
−5
8
.6
8
2
6
0
8
−6
8
.0
4
4
4
1
5
a
4
3
.2
4
8
6
6
5
1
2
.9
7
7
7
1
1
9
0
.5
6
6
7
4
0
6
5
0
.3
7
7
3
1
0
4
8
1
.9
7
5
7
0
8
7
−1
1
.1
3
1
0
9
0
−1
1
.0
6
8
6
3
9
−4
0
.3
6
6
3
1
7
a
5
2
.4
9
2
4
0
3
2
1
.6
0
3
9
3
7
7
−0
.2
0
9
0
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−
t
a
b
l
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4:
P
ar
am
et
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s
fo
r
fi
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to
th
e
d
ow
n
-s
p
in
el
ec
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en
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fo
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fi
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t-
ro
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s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
4.
L
i+
B
e+
B
+
C
+
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+
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+
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+
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e+
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5
1
5
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6
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6
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2
9
9
8
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0
8
9
9
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9
9
2
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2
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2
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6
1
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3
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9
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1
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1
3
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7
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8
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5
4
4
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6
5
0
8
8
2
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6
1
6
2
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7
5
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9
5
0
4
4
3
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0
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2
9
8
2
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1
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1
0
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8
0
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2
9
3
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0
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8
5
1
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4
5
2
2
7
5
2
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4
3
3
4
3
−
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b 8
0
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6
3
1
9
4
4
3
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5
2
2
2
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2
1
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2
4
4
4
6
8
1
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2
7
2
9
2
8
−
−
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b 9
0
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7
5
3
8
2
9
0
0
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8
5
6
3
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3
2
0
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4
5
7
6
3
4
9
0
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6
0
0
9
0
9
2
−
−
−
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0
−
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3
3
1
7
3
6
2
0
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8
0
6
9
6
5
1
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8
8
3
6
5
1
2
−
−
−
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1
−
0
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7
2
7
3
6
7
7
0
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4
6
6
3
1
6
0
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2
−
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8
9
7
7
3
9
3
0
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7
8
1
2
0
3
4
−
−
−
−
−
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3
−
−
−
−
−
−
−
−
t
a
b
l
e
5:
P
ar
am
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er
s
fo
r
fi
ts
to
th
e
u
p
-s
p
in
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p
-s
p
in
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tr
on
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ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
at
om
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
6.
L
i
B
e
B
C
N
O
F
N
e
a
0
6
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0
6
5
9
1
7
3
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2
5
8
9
3
2
1
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0
2
6
7
3
3
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3
8
9
3
2
6
1
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9
6
7
1
6
2
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9
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6
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6
7
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5
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4
6
9
2
8
5
−
1
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8
2
7
7
2
3
2
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5
9
0
5
2
2
−
a
1
3
−
0
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6
3
5
7
0
0
8
−
−
−
2
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8
6
8
0
6
2
1
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9
9
7
8
3
4
−
a
1
4
−
−
−
−
−
0
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3
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1
4
7
7
1
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1
8
4
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−
a
1
5
−
−
−
−
−
−
0
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0
4
5
0
3
5
8
−
b 2
−1
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0
2
8
8
9
0
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8
0
1
1
3
2
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7
0
0
3
4
6
1
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0
6
7
5
3
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9
1
6
6
7
2
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0
7
0
5
9
6
−2
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4
7
9
1
5
9
2
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7
4
8
3
6
1
b 3
1
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9
2
5
4
2
9
2
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2
9
9
7
4
1
0
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6
2
5
4
7
3
0
5
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6
4
5
6
4
4
9
−0
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0
1
0
5
7
7
0
3
3
1
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6
7
2
4
7
7
4
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4
5
9
8
9
9
×
1
0
−
5
0
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0
0
9
7
7
9
1
0
6
7
b 4
0
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5
5
3
5
0
2
3
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5
3
1
6
0
3
1
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8
7
5
0
2
1
2
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3
2
0
9
1
3
1
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9
1
2
9
1
0
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7
8
3
2
2
8
2
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6
2
1
4
2
6
0
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2
3
4
7
3
4
3
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0
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3
9
2
0
5
6
3
1
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8
8
2
3
1
2
2
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4
9
4
6
2
0
1
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5
6
9
0
7
9
1
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2
7
3
7
1
4
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9
5
4
2
9
9
2
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5
9
8
1
4
0
0
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1
7
9
7
3
1
9
b 6
−
1
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5
3
5
8
6
8
−0
.6
1
0
0
4
8
9
1
1
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2
4
3
1
0
5
1
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5
9
3
3
8
9
2
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3
5
7
2
1
4
2
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0
7
5
0
3
7
−
b 7
−
1
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0
6
7
5
4
4
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8
9
9
9
8
3
3
1
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0
6
0
9
7
7
−
1
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7
1
0
4
4
6
1
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4
0
9
4
3
0
−
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−
1
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6
0
2
6
2
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1
2
7
3
1
7
2
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6
7
3
5
9
8
3
−
2
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5
3
2
0
1
0
2
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9
2
1
5
0
0
−
b 9
−
0
.9
1
9
4
9
5
8
5
−
0
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6
1
6
9
8
3
4
−
1
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1
9
4
8
0
2
1
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8
7
5
1
2
7
−
b 1
0
−
0
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1
3
3
8
3
7
9
−
0
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2
7
7
5
7
0
7
−
0
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8
1
0
5
2
3
6
1
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9
5
9
3
0
0
−
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1
−
0
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6
2
7
5
7
7
9
−
−
−
1
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7
1
5
1
8
4
1
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4
4
8
2
1
1
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b 1
2
−
−
−
−
−
0
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1
4
6
3
9
7
7
0
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8
7
2
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3
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1
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b 1
3
−
−
−
−
−
−
0
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7
4
3
8
6
7
7
−
t
a
b
l
e
6:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
u
p
-s
p
in
-d
ow
n
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p
in
el
ec
tr
on
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ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
at
om
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
7.
L
i
B
e
B
C
N
O
F
N
e
a
0
0
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1
5
2
6
4
1
2
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5
4
0
1
6
2
1
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0
7
7
1
1
6
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4
5
4
7
3
2
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8
6
7
8
9
9
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4
6
0
0
2
7
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5
3
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2
8
a
2
5
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8
6
0
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6
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1
1
3
0
2
2
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6
8
7
2
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a
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−
−
−
−
b 2
−0
.5
4
8
6
5
0
5
8
1
.9
4
1
6
3
8
8
0
.7
0
5
1
3
2
8
2
1
.7
4
6
1
3
9
7
1
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1
2
8
3
8
3
2
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3
2
9
8
2
1
2
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6
2
1
2
0
8
3
.0
2
2
5
5
7
8
b 3
2
.2
3
3
9
9
0
0
1
.2
4
7
4
1
1
8
−0
.0
0
1
3
0
6
0
9
3
8
−0
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0
0
7
8
1
4
7
1
9
6
0
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0
2
0
7
5
4
2
4
0
0
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0
6
2
4
1
4
8
3
3
0
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0
1
3
2
4
9
8
8
7
−0
.0
0
6
9
0
4
8
8
6
9
b 4
2
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4
0
9
3
1
3
3
.9
6
2
4
9
6
8
−0
.0
0
2
2
7
6
1
8
2
9
−0
.0
5
8
9
3
9
5
8
2
2
.0
1
1
4
8
2
2
0
.6
1
8
6
4
2
9
8
−4
.3
2
6
5
6
2
2
−6
.0
9
3
7
5
0
1
b 5
2
.1
5
6
2
1
6
8
2
.1
2
5
8
1
4
7
0
.8
5
9
5
5
0
8
8
1
.0
4
1
6
2
1
9
2
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5
2
8
4
0
4
5
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5
7
9
2
9
0
6
.5
0
2
6
3
4
4
9
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5
9
5
6
6
2
b 6
3
.0
8
1
9
7
1
8
−0
.0
1
9
3
6
4
2
2
8
0
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2
8
6
1
1
3
7
2
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8
3
3
8
6
7
1
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0
5
9
3
5
5
3
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8
9
0
0
7
1
2
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6
8
4
9
4
7
4
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7
9
7
4
0
3
b 7
3
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9
9
9
6
7
4
1
.6
3
2
2
5
2
8
−
2
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6
4
5
9
3
6
−
1
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4
1
3
0
0
6
1
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0
6
7
1
3
9
3
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6
3
7
2
3
4
b 8
1
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3
5
0
2
4
9
2
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0
3
5
2
1
5
−
2
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2
8
9
7
5
3
−
1
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6
8
1
7
1
1
1
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5
9
2
9
3
7
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4
6
4
0
6
5
b 9
0
.0
0
0
6
2
3
4
8
4
4
0
1
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3
3
9
3
9
8
−
1
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2
9
4
1
3
2
−
1
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6
4
1
1
8
8
0
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3
4
9
2
0
9
5
2
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1
4
3
1
8
2
b 1
0
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0
8
0
8
4
9
×
1
0
−
5
0
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6
7
6
7
3
7
1
−
0
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7
9
4
0
6
8
2
−
0
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3
7
0
5
5
7
4
−
0
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6
9
7
6
8
7
2
b 1
1
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6
8
5
3
9
6
×
1
0
−
5
0
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5
8
5
2
6
1
5
−
0
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4
3
0
6
5
2
3
−
−
−
1
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6
1
5
9
0
5
b 1
2
0
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6
6
3
6
6
4
0
−
−
1
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8
4
6
7
2
6
−
−
−
0
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3
2
8
2
0
7
0
b 1
3
−
−
−
0
.7
9
8
0
0
4
2
4
−
−
−
−
t
a
b
l
e
7:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
d
ow
n
-s
p
in
-d
ow
n
-s
p
in
el
ec
tr
on
-p
ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
at
om
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
6.
L
i
B
e
B
C
N
O
F
N
e
a
0
−
4
.0
8
1
5
6
5
1
2
.4
4
9
9
1
8
8
1
.4
4
6
1
6
4
1
0
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7
3
5
2
7
5
5
−1
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2
6
9
1
6
0
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3
1
2
4
2
6
−3
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2
0
8
6
1
2
a
2
−
2
2
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5
6
2
4
7
1
2
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5
7
6
4
2
3
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1
1
4
1
6
2
4
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8
2
8
6
9
1
−5
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3
6
4
8
1
7
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1
4
0
5
0
8
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6
0
4
0
2
8
a
3
−
4
0
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5
5
0
1
9
3
3
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5
7
8
8
9
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0
6
6
1
2
5
6
−1
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5
5
8
9
2
0
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1
2
7
9
4
1
−4
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6
4
3
5
2
9
4
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4
8
3
6
0
8
a
4
−
2
0
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6
0
7
0
1
1
1
1
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8
8
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3
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9
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1
2
2
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9
5
3
7
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a
5
−
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1
3
7
1
5
0
1
3
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9
5
3
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5
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7
1
5
2
9
0
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6
8
9
2
1
9
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8
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8
0
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2
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1
1
6
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a
6
−
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3
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6
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8
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9
0
4
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8
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1
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8
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5
2
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3
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a
7
−
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3
5
0
9
4
2
7
1
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4
6
8
2
9
4
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8
6
1
9
1
4
−
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8
1
1
3
9
9
8
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4
1
8
5
5
6
0
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2
4
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0
9
a
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−
2
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5
2
1
5
0
5
1
8
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7
1
3
2
9
8
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7
2
2
2
2
9
−
1
0
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0
3
1
0
9
6
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9
8
3
7
2
5
−
a
9
−
2
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1
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3
8
7
9
3
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8
6
3
6
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6
7
1
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3
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−
8
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6
5
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0
9
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3
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9
8
0
9
2
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a
1
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−
1
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9
5
7
2
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6
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6
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1
2
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3
3
9
1
1
3
−
−0
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2
0
2
4
6
1
3
3
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8
1
9
9
6
3
−
a
1
1
−
0
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8
2
1
8
3
7
8
0
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9
2
0
6
3
4
8
3
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7
5
4
2
4
4
−
2
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6
8
2
1
6
0
3
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8
6
1
4
9
0
−
a
1
2
−
−
1
.7
5
7
2
6
0
7
5
.9
0
9
5
5
9
6
−
0
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3
8
0
5
8
0
0
1
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6
8
3
9
8
1
−
a
1
3
−
−
0
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3
9
1
8
3
3
9
4
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5
6
1
4
0
6
−
−
1
.8
3
2
4
4
3
2
−
a
1
4
−
−
−
2
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3
2
5
5
1
1
−
−
0
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4
7
6
1
1
6
1
−
a
1
5
−
−
−
0
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1
6
4
1
8
6
9
−
−
−
−
b 2
−
−2
.3
8
6
2
1
6
5
−1
.1
8
3
5
7
1
8
0
.8
5
4
0
4
6
6
4
0
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1
7
0
7
8
2
4
−3
.0
3
3
0
2
9
8
2
.4
7
1
5
8
6
1
1
.9
9
3
6
1
9
8
b 3
−
−3
.2
5
3
8
0
3
1
5
.2
6
4
6
7
8
0
0
.0
1
8
1
0
8
4
5
7
−0
.0
0
0
7
0
1
0
9
1
9
1
2
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2
5
7
4
6
8
0
.0
0
1
9
2
8
1
7
6
9
−0
.0
0
2
0
9
3
3
3
0
9
b 4
−
1
.3
3
0
6
5
0
5
4
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8
1
5
5
9
1
1
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5
1
4
3
1
0
−
−3
.2
6
3
9
4
4
6
−2
.6
0
3
0
8
2
8
0
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8
1
7
0
8
4
1
b 5
−
0
.9
7
7
9
5
9
4
5
2
.5
3
8
0
8
0
7
−1
.7
8
8
6
5
8
3
−
0
.0
0
0
7
6
3
4
8
7
0
4
−2
.9
4
7
4
0
6
9
0
.8
6
8
2
0
7
8
2
b 6
−
1
.7
8
1
3
6
7
3
4
.3
8
2
6
1
4
1
−1
.5
6
2
5
8
2
3
−
1
.9
6
3
7
2
0
2
1
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7
9
8
7
0
0
−
b 7
−
1
.3
4
5
8
4
3
2
−2
.8
5
8
6
6
0
7
1
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4
7
9
9
5
8
−
2
.0
4
6
2
3
0
5
1
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3
0
7
3
9
8
−
b 8
−
0
.0
5
5
5
4
4
7
8
8
−1
.7
7
3
0
6
7
4
0
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1
2
3
7
7
0
1
−
0
.5
4
5
6
9
5
3
4
2
.0
8
8
2
5
5
7
−
b 9
−
0
.8
7
6
4
4
1
0
8
0
.7
1
2
0
8
7
2
2
0
.5
3
5
7
7
0
6
8
−
1
.1
0
8
3
4
1
8
1
.2
2
6
1
2
1
6
−
b 1
0
−
−
0
.7
7
0
8
0
5
7
2
1
.3
0
3
8
4
8
5
−
0
.6
2
5
7
5
3
3
0
0
.9
6
7
6
7
0
6
5
−
b 1
1
−
−
0
.7
6
3
4
3
5
3
2
1
.2
2
7
9
8
6
9
−
−
0
.9
8
2
7
3
6
9
9
−
b 1
2
−
−
−
1
.1
1
1
7
6
6
0
−
−
0
.7
6
2
6
7
6
9
5
−
b 1
3
−
−
−
0
.8
7
0
9
7
1
0
6
−
−
−
−
t
a
b
l
e
8:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
u
p
-s
p
in
-u
p
-s
p
in
el
ec
tr
on
-p
ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
io
n
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
6.
L
i+
B
e+
B
+
C
+
N
+
O
+
F
+
N
e+
a
0
−
3
.7
4
7
8
3
1
7
2
.2
1
7
5
8
4
1
−0
.1
5
2
7
0
3
3
0
−1
.6
0
6
0
6
1
1
−2
.7
4
0
4
6
0
7
−3
.4
3
2
4
8
3
9
−3
.9
9
8
0
7
2
8
a
2
−
5
.8
3
6
2
5
1
1
2
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1
9
5
4
7
3
5
.7
2
0
6
3
0
9
2
.4
0
9
2
6
8
7
−0
.6
3
9
5
3
2
6
0
−2
.8
1
9
9
6
7
1
−1
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3
0
4
7
2
0
a
3
−
3
.6
4
1
4
6
9
8
−1
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7
8
2
0
3
9
−3
.3
8
8
6
2
2
3
−3
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6
9
2
1
5
3
−1
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4
6
0
4
5
4
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2
1
6
4
6
8
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1
0
0
8
7
3
a
4
−
8
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1
5
9
6
0
2
1
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7
7
7
9
8
7
1
4
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3
0
7
6
1
2
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6
0
8
5
7
8
4
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5
3
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a
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1
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6
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2
a
6
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8
5
6
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3
−
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6
6
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0
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2
8
2
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8
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9
2
6
7
7
2
2
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5
7
8
2
4
7
4
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6
6
3
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1
a
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3
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4
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8
3
−
−
0
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9
4
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7
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1
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9
3
7
7
7
1
1
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3
8
7
3
9
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3
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9
0
9
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6
1
a
1
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3
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1
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−
−
2
.5
3
4
2
7
3
0
0
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0
9
9
3
5
9
0
2
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1
2
4
9
4
4
3
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0
8
8
4
7
9
a
1
1
−
4
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8
0
3
4
4
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−
−
2
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7
5
5
0
6
4
−
2
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4
1
9
1
2
4
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5
5
3
7
3
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a
1
2
−
3
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1
6
3
2
4
0
−
−
1
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7
3
9
4
5
3
−
1
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9
9
1
4
6
5
2
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6
5
9
2
6
2
a
1
3
−
1
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8
5
3
9
7
5
−
−
1
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3
4
6
2
7
1
−
1
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5
1
0
9
8
5
1
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3
1
3
5
0
6
a
1
4
−
1
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9
1
8
0
6
8
−
−
0
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6
3
6
1
5
9
8
−
0
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1
8
2
0
7
0
4
2
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5
6
2
0
0
9
a
1
5
−
0
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4
0
0
9
8
9
3
−
−
−
−
−
0
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0
1
7
0
5
3
1
b 2
−
1
.0
9
5
5
0
4
1
0
.0
0
0
1
9
1
3
2
3
6
4
2
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0
7
0
3
1
2
1
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6
8
7
7
2
8
2
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3
3
9
5
2
5
2
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9
4
8
8
4
2
2
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3
4
8
0
8
2
b 3
−
1
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0
6
0
7
2
7
0
.7
3
7
5
9
6
7
8
1
.3
8
2
5
3
0
9
0
.0
6
2
9
6
7
7
0
4
0
.0
0
1
4
8
5
9
8
6
9
−0
.0
0
1
0
2
0
7
9
5
3
0
.0
0
0
8
7
6
4
8
0
7
8
b 4
−
1
.5
3
2
2
0
4
5
−
1
.7
2
7
3
0
9
6
2
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3
2
7
3
4
4
1
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0
7
7
9
3
5
2
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8
5
0
8
4
5
1
.0
1
4
3
2
2
3
b 5
−
1
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2
3
9
9
1
1
−
0
.5
1
1
1
9
4
7
0
1
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4
0
7
7
2
7
1
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0
6
1
6
9
3
2
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7
3
1
7
2
9
−3
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5
1
6
9
2
2
b 6
−
1
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7
9
2
9
8
0
−
0
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6
1
1
5
1
2
9
−1
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9
4
5
1
7
6
0
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6
7
1
3
8
0
2
1
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9
7
4
7
8
3
0
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0
9
1
0
4
9
8
b 7
−
1
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4
3
8
2
0
0
−
−
1
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6
8
9
7
7
6
1
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4
6
2
0
0
7
1
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2
1
8
1
2
4
0
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7
8
5
7
0
2
5
b 8
−
1
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3
8
3
1
5
9
−
−
1
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9
6
0
9
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0
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2
3
1
8
3
9
2
1
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8
8
1
1
1
3
2
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5
5
7
7
8
9
b 9
−
1
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8
6
4
7
9
8
−
−
0
.7
1
5
4
1
3
3
9
−
1
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6
5
1
7
8
4
1
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2
1
1
4
5
8
b 1
0
−
0
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9
0
0
3
8
8
5
−
−
0
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2
0
0
4
9
0
3
−
1
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3
9
2
0
0
7
1
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4
6
9
5
0
9
b 1
1
−
0
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6
4
9
5
6
1
4
−
−
1
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7
5
5
8
4
9
−
1
.0
9
1
8
2
4
5
1
.0
2
0
2
7
7
5
b 1
2
−
0
.9
2
3
1
2
3
9
9
−
−
0
.7
3
0
3
8
4
4
1
−
0
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0
2
3
2
2
4
2
1
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5
7
2
3
4
4
b 1
3
−
0
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5
9
8
8
6
6
5
−
−
−
−
−
0
.6
8
9
9
3
9
3
8
t
a
b
l
e
9:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
u
p
-s
p
in
-d
ow
n
-s
p
in
el
ec
tr
on
-p
ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
io
n
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
7.
L
i+
B
e+
B
+
C
+
N
+
O
+
F
+
N
e+
a
0
0
.6
2
5
5
2
9
7
5
−0
.4
4
9
0
9
4
4
5
−1
.2
5
4
5
8
5
7
−1
.8
6
0
5
8
2
1
−2
.4
0
5
6
1
2
3
−2
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5
6
2
7
9
3
−3
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5
5
4
1
2
7
−3
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1
8
2
3
3
5
a
2
7
.2
3
7
3
9
9
7
9
.0
6
8
5
0
1
8
1
1
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5
6
7
0
3
1
9
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5
4
4
2
6
3
0
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1
2
7
4
8
3
7
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3
8
8
5
6
2
6
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2
3
3
7
8
3
4
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5
8
7
8
2
a
3
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4
5
8
1
6
6
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0
4
9
8
9
6
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4
1
8
8
7
6
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6
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1
4
9
4
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2
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8
4
1
4
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6
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9
2
1
1
6
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3
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0
9
3
9
8
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0
4
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7
0
0
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a
4
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1
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2
a
5
6
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3
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5
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1
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1
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1
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9
2
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9
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3
9
9
8
7
6
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7
4
9
8
8
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7
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1
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1
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0
0
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1
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1
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5
2
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4
6
5
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1
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9
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4
1
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8
7
4
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3
7
3
1
8
1
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1
7
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3
2
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8
6
1
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5
9
5
9
8
a
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1
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1
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3
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1
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1
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9
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2
9
7
6
0
9
8
−
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−
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7
1
8
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4
2
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6
6
2
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a
1
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3
0
3
6
5
6
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1
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9
8
6
8
1
2
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3
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2
0
9
0
−
−
−
0
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7
3
7
6
4
1
2
2
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1
2
9
4
8
5
a
1
1
−
1
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7
3
6
4
4
7
4
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8
9
8
2
0
7
−
−
−
1
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8
1
2
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3
2
−
a
1
2
−
0
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3
8
5
5
8
6
4
2
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4
2
8
4
4
7
−
−
−
1
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3
3
9
1
3
9
−
a
1
3
−
−
0
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9
0
1
4
1
7
6
−
−
−
1
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3
0
9
2
0
6
−
a
1
4
−
−
0
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9
5
0
1
9
4
4
−
−
−
−
−
a
1
5
−
−
−
−
−
−
−
−
b 2
0
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8
0
5
9
5
2
6
1
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2
8
0
8
4
4
1
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5
9
8
3
3
0
0
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9
1
9
7
6
6
6
0
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2
1
7
7
1
3
8
1
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7
4
0
9
6
6
2
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3
7
7
0
4
1
2
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4
6
3
6
0
2
b 3
0
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9
6
4
5
0
5
4
0
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0
1
7
9
0
2
6
0
7
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3
0
2
9
5
8
5
0
0
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0
2
4
3
4
5
7
6
6
0
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0
0
4
1
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3
2
6
3
4
0
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0
0
4
4
5
4
8
3
0
1
0
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1
4
6
1
6
0
2
5
0
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0
1
0
4
3
6
6
3
8
b 4
1
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3
4
3
3
9
3
0
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6
8
2
6
5
5
2
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1
8
7
2
4
1
3
0
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6
1
5
4
7
1
5
2
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8
2
5
6
7
6
3
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0
4
4
8
8
8
−2
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7
6
1
0
9
5
6
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7
2
1
0
5
0
b 5
1
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0
1
8
8
1
8
1
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4
0
5
7
6
1
2
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8
6
4
0
6
5
2
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0
5
6
5
8
5
1
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9
6
3
6
9
5
2
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7
7
5
6
3
2
7
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4
1
8
1
0
4
6
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0
0
6
9
2
2
b 6
0
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0
0
6
9
5
9
0
3
2
7
0
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0
5
5
3
0
4
2
2
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7
5
5
3
5
5
0
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6
5
4
9
6
8
4
−
−
3
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1
2
8
4
2
1
−0
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0
3
9
0
2
1
1
6
5
b 7
0
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9
6
0
9
4
5
6
0
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0
3
1
6
4
7
7
3
5
2
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0
4
6
4
2
5
−
−
−
1
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9
8
6
0
3
2
3
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3
9
2
6
9
5
b 8
0
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8
0
2
4
3
8
0
0
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5
2
5
5
9
4
2
2
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1
9
7
1
5
1
−
−
−
2
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1
8
1
3
8
3
1
.7
1
9
0
3
5
8
b 9
−
0
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5
0
8
4
6
5
2
1
.2
1
2
5
6
7
7
−
−
−
1
.8
3
2
9
5
9
3
−
b 1
0
−
1
.0
2
4
8
9
6
4
0
.1
9
0
5
2
1
1
1
−
−
−
0
.8
0
1
2
0
1
1
5
−
b 1
1
−
−
0
.7
9
2
4
4
4
0
7
−
−
−
1
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4
2
2
8
3
7
−
b 1
2
−
−
1
.0
1
7
4
2
5
0
−
−
−
−
−
b 1
3
−
−
−
−
−
−
−
−
t
a
b
l
e
10
:
P
ar
am
et
er
s
fo
r
fi
ts
to
th
e
d
ow
n
-s
p
in
-d
ow
n
-s
p
in
el
ec
tr
on
-p
ai
r
in
tr
ac
u
le
d
en
si
ti
es
fo
r
th
e
fi
rs
t-
ro
w
io
n
s
to
th
e
fo
rm
gi
ve
n
b
y
E
q
.
3.
6.
L
i+
B
e+
B
+
C
+
N
+
O
+
F
+
N
e+
a
0
−
−
2
.2
2
3
5
3
3
3
1
.2
0
6
1
3
3
1
0
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6
5
7
6
8
0
6
−0
.2
6
6
6
6
6
0
9
−2
.4
0
3
3
1
5
7
−3
.5
5
2
9
0
4
0
a
2
−
−
2
.7
7
8
7
7
1
1
3
.7
3
5
9
7
4
8
4
.8
1
1
1
4
7
1
2
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4
6
1
8
7
9
−5
.1
1
2
0
0
4
7
−0
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6
3
5
1
3
6
4
a
3
−
−
−1
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2
9
7
4
1
2
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0
8
6
7
7
1
−2
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4
4
5
2
1
9
3
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9
3
1
7
6
7
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5
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6
3
7
8
3
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2
0
5
3
3
3
6
a
4
−
−
1
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0
0
6
0
5
7
1
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3
1
9
3
9
9
1
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0
0
7
6
6
5
3
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8
9
4
8
8
0
3
9
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0
5
5
6
3
4
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1
2
4
1
7
9
a
5
−
−
0
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8
3
9
8
9
8
9
0
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5
5
1
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8
7
6
0
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0
6
4
6
3
3
9
2
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7
0
8
3
7
9
1
3
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7
5
5
2
1
3
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8
9
3
9
0
1
a
6
−
−
−
−
−
3
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5
6
0
0
9
1
3
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9
4
8
9
2
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3
9
0
1
5
9
8
1
a
7
−
−
−
−
−
0
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9
0
4
5
7
5
9
1
5
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5
5
1
8
3
0
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4
6
5
4
7
1
6
a
8
−
−
−
−
−
−
6
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1
3
0
2
0
2
4
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9
5
4
4
3
4
a
9
−
−
−
−
−
−
2
.1
6
2
7
2
7
9
4
.8
4
3
9
8
9
9
a
1
0
−
−
−
−
−
−
0
.1
0
5
4
1
9
3
7
2
.4
2
6
0
4
2
0
a
1
1
−
−
−
−
−
−
0
.4
8
9
2
3
6
0
1
1
.2
6
6
5
8
6
7
a
1
2
−
−
−
−
−
−
−
2
.0
0
0
8
9
8
1
a
1
3
−
−
−
−
−
−
−
0
.5
8
9
8
2
9
8
3
a
1
4
−
−
−
−
−
−
−
−
a
1
5
−
−
−
−
−
−
−
−
b 2
−
−
0
.1
6
1
7
9
6
3
5
0
.1
0
7
4
7
8
1
6
0
.3
8
0
3
4
7
6
5
−1
.6
8
1
3
6
4
0
2
.8
8
2
3
1
8
6
2
.1
5
9
6
8
0
9
b 3
−
−
0
.5
6
7
9
4
4
2
6
0
.5
2
7
1
9
6
7
5
0
.4
3
1
4
8
2
2
5
0
.1
1
7
5
9
0
8
4
1
.6
1
2
0
7
9
8
−0
.0
0
4
9
9
1
6
8
9
6
b 4
−
−
−
−
−
1
.1
3
9
8
8
2
3
3
.1
5
0
7
5
6
6
1
.1
2
0
5
2
0
9
b 5
−
−
−
−
−
0
.8
5
0
3
9
2
3
7
2
.1
1
3
0
1
5
7
1
.7
0
6
5
4
6
5
b 6
−
−
−
−
−
−
1
.3
1
8
2
8
7
7
1
.5
6
7
7
3
3
8
b 7
−
−
−
−
−
−
1
.7
9
7
3
5
6
4
1
.5
7
9
5
2
2
6
b 8
−
−
−
−
−
−
0
.7
5
0
3
4
9
4
7
1
.1
8
9
4
8
0
9
b 9
−
−
−
−
−
−
0
.0
9
9
5
5
6
0
1
4
1
.0
4
3
3
7
7
9
b 1
0
−
−
−
−
−
−
−
0
.9
7
7
0
2
9
8
2
b 1
1
−
−
−
−
−
−
−
0
.6
9
2
3
2
9
0
2
b 1
2
−
−
−
−
−
−
−
−
b 1
3
−
−
−
−
−
−
−
−