Second-order self-consistent eld algorithms from classical to quantum nuclei

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Second-order self-consistent ﬁeld

algorithms: from classical to quantum

nuclei

Robin Feldmann, Alberto Baiardi, and Markus Reiher∗

ETH Z¨urich, Laboratorium f¨ur Physikalische Chemie,

Vladimir-Prelog-Weg 2, 8093 Z¨urich, Switzerland

December 7, 2022

Abstract

This work presents a general framework for deriving exact and approx-

imate Newton self-consistent ﬁeld (SCF) orbital optimization algorithms

by leveraging concepts borrowed from diﬀerential geometry. Within this

framework, we extend the augmented Roothaan–Hall (ARH) algorithm

to unrestricted electronic and nuclear-electronic calculations. We demon-

strate that ARH yields an excellent compromise between stability and

computational cost for SCF problems that are hard to converge with con-

ventional ﬁrst-order optimization strategies. In the electronic case, we

show that ARH overcomes the slow convergence of orbitals in strongly-

correlated molecules with the example of several iron-sulfur clusters. For

nuclear-electronic calculations, ARH signiﬁcantly enhances the conver-

gence already for small molecules, as demonstrated for a series of proto-

nated water clusters.

∗Corresponding author; e-mail: markus.reiher@phys.chem.ethz.ch

arXiv:2210.10170v2 [physics.chem-ph] 11 Jan 2023

1 Introduction

Processes that are strongly inﬂuenced by nuclear quantum eﬀects are among

the most daunting targets of quantum chemical simulations. One often invokes

the Born–Oppenheimer (BO) approximation1–3to adiabatically separate the

electronic and the nuclear motion. This separation becomes inaccurate as soon

as the coupling between the electrons and nuclei is strong. This inaccuracy

can be cured by including non-adiabatic eﬀects a posteriori, e.g., with pertur-

bation theory.4Alternatively, the full molecular Schr¨odinger equation can be

solved with so-called nuclear-electronic methods, treating nuclei and electrons

on the same footing. In this way, non-adiabatic and nuclear quantum eﬀects

are automatically included.

Explicitly correlated methods yield the most accurate solution of the full

molecular Schr¨odinger equation. They are, however, limited to few-particle

molecules due to the factorial scaling of their computational cost with system

size.5–8A cost-eﬀective alternative is oﬀered by orbital-based nuclear-electronic

approaches.9–14 Although these methods usually do not reach the accuracy

of their explicitly correlated counterparts, they can be routinely applied to

molecules with dozens of electrons and multiple quantum nuclei.15 Orbital-

based nuclear-electronic methods have often been devised by extending algo-

rithms originally designed for electronic problems to nuclear-electronic ones.14

This has led to the emergence of the nuclear-electronic counterparts of many

wave function-16–27 and density functional theory-based28–31 electronic struc-

ture methods. These developments have revealed that some concepts cannot

be straightforwardly transferred from electronic problems to nuclear-electronic

ones. For instance, molecules with a weakly correlated electronic ground state

may display a strongly-correlated nuclear-electronic wave function.20,27 Quan-

tum entanglement measures extracted from nuclear-electronic density matrix

renormalization group (DMRG) calculations can guide this classiﬁcation but

it was shown that in the nuclear- electronic case it is diﬃcult to classify a

molecule unambiguously as strongly or weakly correlated.22,27 Consequently,

the eﬃciency of an algorithm originally ideated for electronic problems may

change drastically when applied to nuclear-electronic problems.

In this work, we address the challenge that optimizing nuclear-electronic

orbitals with self-consistent ﬁeld (SCF) methods is signiﬁcantly more diﬃcult

than for their purely electronic counterparts. The convergence behavior of the

nuclear-electronic Hartree–Fock optimization algorithms has received little at-

tention in the literature so far, with the exceptions of Refs. 32 and 33. As

in electronic-structure theory, the Roothaan–Hall (RH) diagonalization method

with level-shifting,34 damping,35 and the direct inversion of the iterative sub-

space (DIIS),36,37 represent the state of the art for nuclear-electronic calcu-

lations. Recently,33 a new DIIS variant was developed for nuclear-electronic

methods that simultaneously optimizes the nuclear and electronic coeﬃcients

of the wave function. This algorithm currently represents the most eﬃcient

nuclear-electronic orbital optimization scheme. However, as ﬁrst-order opti-

mization methods, they can converge either slowly, possibly to saddle points,

or even diverge. In electronic structure theory, these issues are especially rel-

evant for strongly correlated systems.38 They can be partially alleviated, e.g.,

by orbital steering.39

The deﬁciencies of the RH approach can be overcome with Newton meth-

ods which, however, require the construction of the orbital Hessian. This step

represents the main bottleneck of the orbital optimization algorithm.40–50 For

restricted Hartree–Fock calculations, this computational bottleneck can be over-

come with the augmented Roothaan–Hall (ARH) method.51,52 ARH is a quasi-

Newton approximate second-order method that iteratively constructs the Hes-

sian by expanding it in the basis of density matrices obtained in previous iter-

ation steps.

Here, we introduce a generic framework for Newton and quasi-Newton op-

timization algorithms for electronic and nuclear-electronic Hartree–Fock. We

derive the Newton equations by leveraging diﬀerential geometry techniques53

and extend the ARH method to unrestricted-electronic and nuclear-electronic

problems within this framework. We ﬁrst demonstrate the eﬃciency of our new

implementation for the unrestricted electronic case with the example of iron-

sulfur clusters. The electronic wave function of these systems is known to display

strong correlation eﬀects.54 This renders the solution of the SCF equations very

challenging even with sophisticated convergence acceleration techniques. Sub-

sequently, we compare diﬀerent optimization algorithms for nuclear-electronic

Hartree–Fock calculations. We choose as test cases water clusters because they

are known to exhibit strong nuclear quantum eﬀects.15,55–59 We compare the

eﬃciency of our new ARH method with the geometric direct minimization

(GDM)60,61 and DIIS methods, as well as with a newly implemented exact

Newton method. We show that ARH is signiﬁcantly faster than GDM and

more robust than DIIS with a limited computational overhead compared to the

much more demanding full Newton optimization.

The work is organized as follows: We introduce in Sec. 2the Newton method

from a diﬀerential geometry perspective and show how to apply it to restricted

Hartree–Fock theory. We then extend the Newton method to unrestricted and

nuclear-electronic problems and derive the corresponding exact Newton and

ARH working equations in Sec. 3. Afterwards, in Sec. 4we review the trust-

region augmented Hessian approach which we apply to ARH to ensure that the

algorithm converges to a minimum of the energy functional. After presenting

the computational details in Sec. 5, we discuss the results for electronic and

nuclear-electronic systems in Sec. 6.

2 A diﬀerential geometry perspective on Hartree–

Fock theory

We start this section with a brief account of our formalism for the restricted elec-

tronic BO Hartree–Fock method. We then describe the Hartree–Fock optimiza-

tion from a diﬀerential geometry perspective and derive the Newton equations

for the restricted Hartree–Fock method. Although often overlooked, diﬀeren-

tial geometry is a powerful tool for studying optimization problems in quantum

chemistry.62–65 Subsequently, we show how to enhance computational eﬃciency

of the solution of the Newton equations.

2.1 Restricted Hartree–Fock theory

In Hartree–Fock theory, the N-electron Hartree–Fock wave function, ΨBO, is

taken as a Slater determinant, Φ deﬁned as an antisymmetrized product of

spin-orbitals, φis,

ΨBO(r) = Φ(r) = 1

√N!S− S

φis(ris))!.(1)

Here, ris a vector collecting the positions of all electrons, ris is the coordinate

of the i-th electron with spin quantum number Sand spin projection onto the

z-axis s=−S, −S+ 1, . . . , S.Nsis the number of electrons with a given

spin. Although for electrons the total spin is S=1

2, we derive the equations

for a generic Sto facilitate their extension to the nuclear-electronic case. The

antisymmetrization operator, S−, enforces the correct permutational symmetry

of the wave function. The spin-orbitals φis are expressed as a linear combination

of NAO pre-deﬁned Gaussian-type orbitals {χµ(r)}NAO

µ=1 , which are referred to as

atomic orbitals,

φis(ris) =

NAO

µ=1

Cs,iµχµ(ris).(2)

The coeﬃcients Cs,iµ of the orbitals with spin scan be grouped into a matrix,

Cs, which can be further partitioned into an occupied block, Cs,o, of dimension

NAO ×Ns, and a virtual block, Cs,v, of dimension NAO ×(NAO −Ns). Only

the occupied orbitals enter the Hartree–Fock wave function. Therefore, its cor-

responding energy is completely determined by the AO density matrices, Ds,

which are deﬁned as

Ds=Cs,oCT

s,o.(3)

The density matrices satisfy three conditions, namely idempotency, symmetry

with respect to transposition, and the trace being equal to Ns:

s=Ds=DT

s,Tr (Ds) = Ns.(4)

The electronic Hartree–Fock energy can be expressed as

EBO({Ds}) =

NAO

µν

hµν Dsνµ+1

NAO

µνρσ

Dsνµ ¯

Vµν,ρσDsσρ+

s<t

NAO

µνρσ

DsνµVµν,ρσDtσρ ,

(5)

where hµν contains the one-electron integrals, Vµν,ρσ the Coulomb integrals for

basis functions µ, ν of the ﬁrst electron and ρ, σ of the second one, and ¯

Vµν,ρσ

denotes the antisymmetrized Coulomb integral. Electrons are spin-1

2Fermions

for which the allowed values of the spin projection sare 1

2and −1

2, which we

will denote as ↑and ↓, respectively. In the restricted electronic Hartree–Fock

method, the spatial part of the spin-orbitals are equal, i.e., C↑=C↓=Cand,

therefore, D↑=D↓=D. This reduces the number of free parameters and

simpliﬁes Eq. (5) to

EBO(D) = 2

NAO

µν

hµν Dνµ +1

NAO

µνρσ

Dνµ(2Vµν,ρσ −Vµσ,ρν )Dσρ.(6)

The restricted Hartree–Fock method optimizes the energy, Eq. (6), with re-

spect to the orbital coeﬃcients under the constraint that the orbitals remain

orthonormal

minnE(Co)Co∈RNAO×N/2,CT

oCo=o.(7)

where Cois the occupied block of C. The energy minimization can be equiv-

alently carried out with respect to the density matrix D. In this case, the

orthogonality-constrained optimization problem reads

minnE(D)D∈RNAO×NAO ,D2=D=DT,Tr (D) = N/2.o.(8)

In the following, we demonstrate how to solve these minimization problems

by leveraging diﬀerential geometry. This mathematical discipline provides an

elegant and convenient framework to derive eﬃcient optimization algorithms for

functions of matrix arguments with constraints on the parameters.53

2.2 Optimizing the energy functional on a manifold

The Newton method is the standard second-order method to optimize iteratively

a real-valued function f(x) with x∈Rn, that is at least twice diﬀerentiable.

Starting from an initial guess x0, the function is expanded at the i-th iteration

around the corresponding approximate solution, xi, as

f(xi+h) = f(xi) + gradf(xi)Th+1

2hTHessf(xi)h+O(h3).(9)

Here, Hessf(xi) and gradf(xi) refer to the Hessian and gradient of fat point

xi, respectively. Setting the gradient of Eq. (9) with respect to hto zero yields

the Newton equation:

Hessf(xi)h=−gradf(xi).(10)

The updated approximate solution xi+1 is obtained from the solution of Eq. (10)

xi+1 =xi+h.(11)

This algorithm converges to the stationary point closest to the starting guess

x0.53

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Second-orderself-consistenteldalgorithms:fromclassicaltoquantumnucleiRobinFeldmann,AlbertoBaiardi,andMarkusReiher*ETHZurich,LaboratoriumfurPhysikalischeChemie,Vladimir-Prelog-Weg2,8093Zurich,SwitzerlandDecember7,2022AbstractThisworkpresentsageneralframeworkforderivingexactandapprox-imateNewtonse...

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Second-order self-consistent eld algorithms from classical to quantum nuclei

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