Wave Functions

Wave functions for quantum Monte Carlo are usually called trial functions in VMC and guide functions in DMC, but as far as this work is considered this difference is only in terminology and not in mathematical structure which is always

Ψ = eUX

i

aiΦ(SD)_i . (2.139)

eU _{is called the correlation or the Jastrow part including the Jastrow function U discussed}

below.81

The other factor, P

iaiΦ(SD)i , represents the wave function obtained in any ab initio or

DFT calculation. “SD” stands for Slater determinant, whereas the linear combination shall indicate that this can be either one or many in the sense of single- or multi-configuration wave functions. The discussion of this determinantal part of the guide or trial wave function is the discussion of the various first principles methods presented previously (sections 2.1 and 2.2). Therefore static electron correlation is no problem in QMC as far as it is in the chosen ab initio approach. Note that when dynamic electron correlation is treated implicitly as in DFT, then the wave function obtained is again correlation-free and the need for an explicit treatment of dynamic electron correlation in quantum Monte Carlo is unavoidable.

Typically the QMC guide function’s Jastrow part accounts for the dynamic electron correlation of a many electron system not only between two electrons, but also between one or two electrons and the nucleus. eU is set up to fulfill the Coulomb cusp conditions required for a proper physical description of the system, for example the one specialized for electron-electron singularities proven by Kato:82

∂ ˆΨ ∂rij ! rij=0 = 1 2Ψ for ri+ rj 6= 0. (2.140) ˆ

Ψ in the derivative shall indicate the spherical average is taken over r12= const.

A general form of the cusp condition that does not only hold for the special cases covered by Kato’s form, was derived by Pack and Byers Brown, who used an expansion of the wave function near the coalescence in terms of spherical harmonics.35,83

The typical approaches for trial functions differ only by the exponent U which can have different structures, e.g. a Pad´e form or that of Schmidt and Moskowitz.84,85

The Jastrow exponent used in this work was of the Schmidt-Moskowitz type and thus basing on the transcorrelated method of Boys and Handy.85 It is given by U =P

with UAij = NA X k ∆(lkA, mkA)ckA(¯rAilkAr¯ mkA Aj + ¯r lkA Aj¯r mkA Ai )¯r nkA ij . (2.141)

The ¯r terms are referred to as scaled distances and are defined by ¯ rAi= bArAi 1 + bArAi and ¯rij = dArij 1 + dArij (2.142) containing the non-linear Jastrow terms bA, dA, and

∆(lkA, mkA) =    1 for lkA6= mkA 2 for lkA= mkA. (2.143)

The ckA are called linear Jatrow parameters.

The interactions to describe can be chosen by using different combinations of non- negative l, m, and n. Therefore electron-electron interactions are described when l = m = 0 and n > 0 is introduced in (2.141) leaving a sum that depends only on the electron electron distances UAij = NA X k 4ckA¯rnijkA. (2.144)

Similarly the electron electron distances are omitted, when n = 0 is prescribed. To gain an electron nucleus distance dependence, then only either l or m must differ from zero. The three particle electron electron nucleus interactions are obtained, when only either l or m equals zero. They are also referred to as backflow terms.85

The parameters to describe the strength and range of the distinct correlation contri- butions are the non-linear bA and dA terms and the linear ckA. Those terms have to be

optimized before performing the actual QMC calculation. The outcome of this optimization is easy to rationalize for Kato’s linear electron electron cusp condition. The derivative of the guide function with respect to rij and close to the singularity rij = 0 will be

∂ ˆΨG ∂rij ! rij=0 = ∂e UP iaiΦ(SD)_i ∂rij ! rij=0 = (X n cndAnr(n−1)ij · e UX i aiΦ(SD)i )rij=0 = c1dAeU X i aiΦ (SD) i = c1dAΨG (2.145)

using l = m = 0 and ¯rij = dArij/1 + dArij ≈ dArij. The derivative of P_iaiΦ(SD)_i is zero,

because the term does not depend on electron electron distances rij. c1dA = 1/2 must be

c1 = 1/2. Similarly the other parameters cn are obtained from the derivatives with respect

to the other distances and for other l, m, and n.

In this work the first optimization steps are typically carried out to obtain the linear parameters ckA. These are as many as the types of cusps to describe. Here four electron

electron terms ranging from linear to the power of four are set by n = 1 − 4 and furthermore for every type of atom three electron nucleus terms with l = 2 − 4 or m = 2 − 4 as well as two backflow terms n = l = 2 and n = m = 4. This is then called a 9-term, 14-term etc. Jastrow for one, two, etc. different atoms. A final adjustment of the linear parameters follows. These are as many bA as different nuclei plus one dA. Initially they are all set to 1.0.

The optimization itself is performed within a VMC calculation with rather few blocks and many random walkers, i.e. a representative sample of Ψ2_T. Therefore typical optimizations techniques (Gauß-Newton, Levenberg-Marquardt, etc.) can be employed.85,86 _{The order of}

optimizing the parameters bA, dA, and ckA as well as the inclusion of the linear coefficients

of the Slater determinant (ai) and of the MOs are discussed in literature.87–89 A popular

disagreement between two groups of quantum Monte Carlo people is about the minimum towards which is to optimize. The ones are supporter of the minimization towards the energy, the others stand in for the variance minimization.90Reasons for the former are that the energy is indeed the quantity that should be as low as possible in accordance with the Rayleigh-Ritz quotient. But the energy value at the minimum is unknown. Therefore the variance of the local energy EL[ΨT] compared to the state’s energy ET[ΨT] should be minimized, since it

is necessarily zero for an eigenstate of the Hamiltonian.84 Variance minimization is not as robust and successful as energy minimization, but it is technically easier and therefore used as standard procedure.

Cusp correction

The last modification done to the wave function is the “cusp correction” accounting for the case when an electron approaches the nucleus. This case is represented by the other condition evolved by Kato, so the electron nucleus singularity is given by

∂ ˆΨ ∂rA

!

rA=0

= ZAΨ for rj 6= 0 (2.146)

for the nucleus being located in the origin and having the nuclear charge of ZA.82 Using

Slater type basis functions this condition would automatically be fulfilled, but common ab initio codes do not include these functions, but Gaussian type functions. However, in QMC

they cause fluctuation in the local energy close to the nuclei because they do not cancel the singularity from the potential energy at the nucleus.

Manten and L¨uchow addressed this well-known drawback by introducing a correction algorithm applied to the 1s and 2s basis functions of cusp-free contracted Gaussian-type basis functions.17,91 A Slater function f (r) = ae−αr+ c is fitted to the considered orbitals in the core region (r < 0.2 bohr) in a least-square procedure. Conditions for this fit are the identity of the original and fit function and their second derivatives at the critical radius, where the transition between both functions takes place. To smoothen the transition both functions are connected by an interpolation polynomial in a transition interval of 0.001 bohr.

Since the evaluation of the atomic orbitals and its first and second derivatives is com- putationally costly the function and its derivatives are interpolated using cubic spline. This leads to a larger fluctuation in the functions but accelerates the calculation considerably.86,91