# Reduced explicitly correlated Hartree-Fock approach within the nuclear-electronic orbital framework: theoretical formulation.

Reduced explicitly correlated Hartree-Fock approach within the nuclear-electronic orbital framework: theoretical formulation. Sirjoosingh,A | |
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Title | Reduced explicitly correlated Hartree-Fock approach within the nuclear-electronic orbital framework: theoretical formulation. |

Authors | A Sirjoosingh,MV Pak,C Swalina,S Hammes-Schiffer |

Journal | The Journal of chemical physics |

Issue | 3 |

Issn | 1089-7690 |

Isbn | |

Doi | 10.1063/1.4812257 |

PMID | 23883005 |

Volume | 139 |

Pages | 034102 |

Keywords | |

Website | [[Website::[1]]] |

Publication Year | Jul 2013 |

## Contents

- 1 Abstract
- 2 Acronyms
- 3 I. INTRODUCTION
- 4 II. THEORY
- 5 B. Energy
- 6 where
- 7 where
- 8 where
- 9 where
- 10 C. Fock operators
- 11 D. Spatial Fock operators
- 12 E. Modified Hartree-Fock-Roothaan expressions
- 13 F. Orthogonalization Scheme
- 14 III. DISCUSSION AND PRACTICAL CONSIDERATIONS
- 15 IV . CONCLUSIONS
- 16 APPENDIX A: RXCHF-fe FOCK EXPRESSIONS
- 17 1. Spin orbital basis
- 18 3. Atomic orbital basis
- 19 APPENDIX B: RXCHF-ne FOCK EXPRESSIONS
- 20 1. Spin orbital basis
- 21 2. Spatial orbital basis
- 22 3. Atomic orbital basis
- 23 2. Spatial orbital basis
- 24 APPENDIX C: RXCHF-ae FOCK EXPRESSIONS
- 25 3. Atomic orbital basis
- 26 Tables

## Abstract

The nuclear-electronic orbital (NEO) method treats electrons and select nuclei quantum mechanically on the same level to extend beyond the Born-Oppenheimer approximation. Electron-nucleus dynamical correlation has been found to be highly significant due to the attractive Coulomb interaction. The explicitly correlated Hartree-Fock (NEO-XCHF) approach includes explicit electron-nucleus correlation with Gaussian-type geminal functions during the variational optimization of the nuclear-electronic wavefunction. Although accurate for small model systems, the NEO-XCHF method is computationally impractical for larger chemical systems. In this paper, we develop the reduced explicitly correlated Hartree-Fock approach, denoted NEO-RXCHF, where only select electronic orbitals are explicitly correlated to the nuclear orbitals. By explicitly correlating only the relevant electronic orbitals to the nuclear orbitals, the NEO-RXCHF approach avoids problems that can arise when all electronic orbitals are explicitly correlated to the nuclear orbitals in the same manner. We examine three different NEO-RXCHF methods that differ in the treatment of the exchange between the geminal-coupled electronic orbitals and the other electronic orbitals: NEO-RXCHF-fe is fully antisymmetric with respect to exchange of all electronic coordinates and includes all electronic exchange terms; NEO-RXCHF-ne neglects the exchange between the geminal-coupled electronic orbitals and the other electronic orbitals; and NEO-RXCHF-ae includes approximate exchange terms between the geminal-coupled electronic orbitals and the other electronic orbitals. The latter two NEO-RXCHF methods offer substantial computational savings over the NEO-XCHF approach. The NEO-RXCHF approach is applicable to a wide range of chemical systems that exhibit non-Born-Oppenheimer effects between electrons and nuclei, as well as positron-containing molecular systems.

## Acronyms

Acronyms |

p, 1 => p χ 1 e ∣ ∣ 1 |

p, i => hp(p) is the core nuclear operator , and V ep |

χ 1 e |

x e N => x e 1 )· ·· χ N e |

1 => (1) 2 ≡ ˜ |

AO => atomic orbital |

DFT => density functional theory |

FCI => full configuration interaction |

GTG => Gaussian-type geminal |

HF => Hartree-Fock |

NEO => nuclear - electronic orbital |

OCBSE => orthogonality constrained basis set expansion |

RHF => restricted Hartree-Fock |

RI => resolution of identity |

SCF => self-consistent-field |

UHF => unrestricted Hartree-Fock |

i => iteration, { ψ a |

i − 1 => (i−1) as the geminal-coupled electronic orbital obtained from the |

i−1 => (i − 1)th iteration and { ψ a |

## I. INTRODUCTION

- inclusion nuclear quantum effects electronic structure calculations study variety chemical systems, involving hydrogen transfer significant hydrogen-bonding interactions.
- 1 -- 3 , effects between electrons and nuclei significant, case with proton-coupled electron transfer reactions.
- 4 , 5 account effects, methods not invoke Born-Oppenheimer separation between electrons and specified nuclei developed.
- 6 -- 27 One approach nuclear - electronic orbital (NEO) method, electrons and select nuclei quantum mechanically formalism.
- 20 -- 28 this approach, all electrons and one a few protons are treated quantum mechanically .

- methods developed within the NEO framework based Hartree-Fock (HF) reference, electronproton correlation incorporated analogs of the methods describing electron-electron correlation electronic structure theory .
- 20 The nuclear densities found lack electron-proton correlation, 23 , 29 -- 31 significant electron-proton Coulomb -

- teraction.
- Extensions of the NEO-HF method, second-order perturbation theory (NEO-MP2), 22 determined capturing dynamical correlation.
- principle, full configuration interaction (FCI) wavefunction built NEO-HF reference complete basis set chemical systems.

- More , explicitly correlated methods within the NEO framework developed.
- 23 , 24 , 26 family methods, termed NEO-XCHF ( correlated Hartree-Fock) correlation between electrons and quantum nuclei Gaussian-type geminal (GTG) functions.
- Two correlated wavefunction-based methods (NEO-XCHF 24 NEO-XCHF226) developed shown perform model systems.
- these methods limited terms computational tractability robustness approximations.
- NEO-XCHF requires the evaluation five - particle integrals quantum nucleus, NEO-XCHF2 offers significant advantages, energy contributions approximated.
- Note variety explicitly correlated methods developed description electron-electron correlation electronic structure community .
- 32 -- 48 As discussed , 23 , 24 ma jority approaches not electron-proton correlation

- difference between electron-electron electron-proton correlation arising electron-proton interaction.
- field NEO-HF wavefunction not reference treatments overlocalization nuclear densities.
- electron-proton correlation included selfconsistent-field procedure optimizing orbitals rather than NEO-XCHF NEO-XCHF2 methods.
- 24 , 26

- approach development density functional theory (DFT) formalism within the NEO framework.
- 25 , 27 , 28 NEO-DFT method required the development electron-proton correlation functionals based pair density expressions arising correlated NEO-XCHF wavefunctions.
- this approach, exchange-correlation functionals be used to describe electron-electron correlation junction more electronproton correlation functionals.
- evaluation of the electron-proton correlation energy NEO-DFT calculations larger chemical systems.

- this paper , develop reduced correlated Hartree-Fock method, denoted NEO-RXCHF , within the NEO framework.
- Similar NEO-XCHF , NEO-RXCHF wavefunction-based approach correlation GTG functions.
- difference between two methods RXCHF , only select electronic orbitals are explicitly correlated to the nuclear orbitals, XCHF , all electronic orbitals are explicitly correlated to the nuclear orbitals.
- physical assumption RXCHF only electronic orbitals (e.g., valence orbitals) need to the quantum nuclei, others (e.g., core orbitals) need not correlated.
- enforcing correlation all electrons to the quantum nuclei same manner , XCHF , more than neglecting portions correlation expected , RXCHF .

- consider three different methods within the NEO-RXCHF formalism.
- first method based ansatz nuclear - electronic wavefunction includes coupling only select electronic orbitals to the nuclear orbitals antisymmetric with respect to exchange coordinates.
- this case, antisymmetrization wavefunction results in computational expense similar XCHF .
- second method, electronic orbitals to the nuclear orbitals distinguished other electronic orbitals Hartree product formalism between two types electronic orbitals.
- this case, wavefunction no longer antisymmetric with respect to exchange all coordinates, electronic exchange terms neglected, distinction results in significant improvement tractability .
- third method extension of the second method, including approximate exchange terms between geminal-coupled electrons and the other electrons.
- Each of the first two methods derived ansatz for the wavefunction, third method cannot derived .
- computational expense second

- third methods similar lower than first method.
- remainder of the paper , three methods denoted RXCHF-fe (full exchange), RXCHF-ne (neglect exchange), RXCHF-ae (approximate exchange).

- remainder of the paper , NEO-RXCHF procedure one quantum nucleus one geminalcoupled electronic orbital.
- This approach positronic systems, where all electrons and positron (rather than one nucleus) are treated quantum mechanically , electronic orbital is explicitly correlated to the orbital.
- application NEO-RXCHF positron-containing systems reported accompanying paper .
- 49 systems in which all electrons and hydrogen nucleus are treated quantum mechanically , , expect least two electronic orbitals must be explicitly correlated to the quantum nuclear orbital.
- extension geminal-coupled electronic orbitals is currently under development.
- this paper , retain terminology quantum nucleus, formalism quantum positron.
- each of the first two methods, ansatz for the nuclear - electronic wavefunction derive energy expressions.
- formulate modified Fock operators required analog of the Hartree-Fock-Roothaan procedure for all three methods discuss considerations.

## II. THEORY

- consider system N electrons, one quantum nucleus, assumed proton simplicity , N c fixed classical nuclei.
- Hamiltonian units system given by

- r ,r p , r c denote coordinates electrons, quantum proton, classical nuclei, ,m p mass proton, Z A charge Ath classical nucleus.
- Note term corresponding repulsion between classical nuclei omitted from Eq. (1) simplicity .

- NEO-XCHF approach, 23 , 24 nuclear - electronic wavefunction is assumed to be of the form

- ( x e) =| χ 1 e ( x e 1 )· ·· χ N e ( x N ) Slater determinant N electronic spin orbitals, χ p nuclear spin orbital,

- these expressions and those that follow,x x p denote spin coordinates electrons and the quantum proton, .
- NEO-XCHF energy modified Hartree-Fock equations derived presented in Ref. 24 .
- this ansatz been used to describe electron-electron correlation.
- 48

- NEO-RXCHF approach, nuclear - electronic wavefunction is assumed to be of the form

- This ansatz contains only one electronic spin orbital that is geminal-coupled to the nuclear spin orbital, wavefunction antisymmetric with respect exchange of all electronic coordinates.
- We denote this ansatz as RXCHF-fe to emphasize that this approach includes full exchange.
- Note XCHF and RXCHF-fe wavefunctions different geminal factor is of the form 1 +G XCHF and G RXCHF-fe.
- reason choice discussed accompanying paper 49 positronic systems.
- emphasize interchange between two forms geminal factor by choosing parameters GTG functions.

- This ansatz contains only one electronic spin orbital that is geminal-coupled to the nuclear spin orbital.
- differs fully antisymmetric ansatz Eq. (5) one electron ( coordinate x 1 ) distinguished others.
- electron the `` electron and the other electrons r
*electrons.* - We denote this ansatz as RXCHF-ne to emphasize that this approach neglects exchange between special and regular electrons.

- energies associated with nuclear - electronic wavefunctions defined expectation values operator given in Eq. (1) .

- from Eq. (2) , (5) , (6) generate XCHF , RXCHF-fe, RXCHF-ne energy , .
- Variation Eq. (7) respect to the spin orbitals leads modified Hartree-Fock (HF) equations

- HF method, NEO-XCHF method, electronic Fock operators all same.
- contrast, RXCHF methods, variation energy with respect to the geminal-coupled electronic orbital, χ 1 , not equivalent to variation respect one other electronic orbitals.
- RXCHF methods require the solution three equations

- (1) electronic HF equation in Eq. (8) i= 1, denote 1 e1 ; (2) electronic HF equation in Eq. (8) i= 1, denote 2 =···= N ; (3) HF equation in Eq. (9) .

- Subsections II A -- II E , evaluation of the energy expressions in Eq. (7) RXCHF-fe and RXCHF-ne methods, RXCHF-ae method.
- present unrestricted Hartree-Fock (UHF) formalism report working expressions for the RXCHF methods.

- overlap RXCHF-ne wavefunction evaluated RXCHF - ne RXCHF - ne RXCHF -ne = p ∣ ∣ 2 ∣ ∣ p

- these expressions and those that follow , brackets denote integration over all coordinates, spin-coordinate dependence of the orbitals p x p i x i ( coordinates geminal functions), g g(1, p), coordinate dependence of the spin orbitals bras kets follows order p ,

## B. Energy

- energy associated with the RXCHF-fe wavefunction can be expressed as

## where

- this expression and those that follow , (i) is the core operator ith electron, hp(p) is the core nuclear operator , and V ep ( p, i )= r p r i 1 Coulomb attraction operator between the quantum nucleus and ith electron.

- Development RXCHF-fe formalism general case four more electrons left studies.
- energy associated with the RXCHF-ne wavefunction can be expressed as

- core electronic energy for the regular electrons

- represents usual electronic Coulomb and exchange terms regular electrons.
- contributions defined E G1 = S RXCHF 1 - ne χ p χ 1 e ∣ ∣ 1 ( p, 1) χ p χ 1 e , (27) 1 given by Eq. (14) ,

- contrast to the RXCHF-fe formulation, RXCHF-ne energy expression not include restricted summations.
- formulation of the RXCHF-ne method systems any number electrons.

## where

- overlap given in Eq. (10) .
- omit details of the derivation of the energy components include example derivation of the one-electron RXCHF-fe energy supplementary material.
- 58

- Eq. (21) contains restricted summations, Fock operators all regular electrons not identical, density matrices need solve HF equation .
- remainder this paper , assume RXCHF-fe formulation, system interest contains most three electrons energy expression in Eq. (12) reduces

## where

- this expression and those that follow,V ee ( i, j ) =| r i r j 1 Coulomb repulsion operator between the ith jth electrons.
- third-order terms given by

- By comparing Eqs.
- (12) (24) , that the RXCHF-ne method offers significant advantages in computational tractability RXCHF-fe method.
- by distinguishing one electron geminal-coupled to the quantum nucleus other electrons, RXCHF-ne energy requires the evaluation only up to three-particle integrals, energy associated with the fully antisymmetric ansatz RXCHF-fe requires evaluation of up five-particle integrals, is the case NEO-XCHF .

- motivation maintaining RXCHF-ne level tractability accounting exchange effects between special and regular electrons, propose RXCHF-ae method, includes approximate exchange.
- method, energy expressed

## where

- Note energy contribution part of the second-order energy RXCHF-fe expression given in Eq. (15) .
- RXCHF-ae method, energy in Eq. (30) is minimized with respect to the spin orbitals.
- addition E , , RXCHF-ae energy not arise wavefunction, approximations need invoked evaluate expectation values other operators.
- shown accompanying paper , 49 given set of orbital coefficients, approximate exchange term given in Eq. (31) accounts more than 99% difference RXCHF-fe and RXCHF-ne energies for the positron-lithium system, where all electrons and positron are treated quantum mechanically .

## C. Fock operators

- three modified HF equations corresponding to the nuclear orbital, geminal-coupled electronic orbital, regular electronic orbitals solved self-consistently procedure.
- three spin-coordinate-dependent equations

- Note Eq. (34) includes terms off-diagonal Lagrange multipliers ensure geminal-coupled orbital is constrained to be orthogonal regular electronic orbitals.
- terms eliminated orthogonalization scheme Sec.
- II F .

- Fock operators equations determined methods by varying the energy expressions

- Eqs.
- (23) , (24) , (30) respect to the orbitals.
- resulting expressions for RXCHF-fe, RXCHF-ne, RXCHF-ae are given in Section 1 Appendices , B , C , .

## D. Spatial Fock operators

- To obtain the spatial Fock operators eigenvalue equations, integrate Eqs.
- (33) -- (35) over the spins electrons spin of the quantum nucleus.
- all RXCHF methods, spatial part of the quantum nuclear spin orbital denoted ψ p without specifying spin of this particle integration over the spin this orbital leads unity expectation value expressions spin-independent operators.
- without loss generality , assume spin of the geminal-coupled orbital,

- ψ 1 spatial part of the geminal-coupled spin orbital.
- following unrestricted Hartree-Fock (UHF)

- | | =N α |B| =N β N α +N β =N−1.
- Note index spans different range context spatial orbitals ψ α , 1≤ ≤N α , ψ β ,

- integration of Eqs.
- (33) (34) over spin results in spatial-coordinate analogs.
- integration of Eq. (35) α β spin leads two different sets

- UHF formalism applied RXCHF methods requires the solution four modified HF equations.
- resulting spatial Fock operator expressions for RXCHF-fe, RXCHF-ne, RXCHF-ae are given in Section 2 Appendices , B , C , .
- Note approach lead spin contamination.
- address issue, developing open-shell formalism ROHF methods electronic structure theory .

## E. Modified Hartree-Fock-Roothaan expressions

- define atomic orbital (AO) basis sets report analogs Hartree-Fock-Roothaan equations for the RXCHF methods.
- expand spatial orbital set N pbf basis functions

- most general case, expand all spatial orbitals set N ebf basis functions.

- Note substantial computational savings gained by restricting basis set geminal-coupled electronic orbital.
- Utilizing knowledge chemical environment quantum nucleus, only subset AO basis functions centered atoms included expansion Eq. (41) .
- example, hydrogen-bonding interface hydrogen nucleus treated quantum mechanically well-described by geminal-coupled electronic orbitals comprised AOs only donor , acceptor , hydrogen atoms interface.

- spatial Fock operators discussed Subsection II D defined terms these quantities provide working expressions form FC = SCE modified Fock matrices F AO basis.
- expressions for the RXCHF-fe, RXCHF-ne, RXCHF-ae Fock matrices are given in Section 3 Appendices , B , C , .

## F. Orthogonalization Scheme

- RXCHF schemes described , modified HF equation Fock operator e1 geminal-coupled electronic orbital solved modified HF equations for the regular alpha beta electronic orbitals Fock operators fα fβ , .
- iteration self-consistent-field (SCF) procedure, geminal-coupled electronic orbital, χ 1 , assumed spin α , not regular alpha electronic orbitals, χ A. ( spin orthogonality,χ 1 all beta electronic orbitals, χ B.) ensure ψ 1 remains spatial part of the regular alpha electronic orbitals, adopt modification orthogonality constrained basis set expansion (OCBSE) method.
- 50

- formulation of the OCBSE method applied open-shell HF schemes, one set orbitals is constrained to be orthogonal set by expanding basis set orthogonal complement of the spaces.
- 50 , 51 The coupled HF equations projected basis sets order eliminate off-diagonal Lagrange multipliers allow use matrix equations SCF procedure.
- drawback OCBSE procedure solutions coupled HF equations calculated more space than be the case expanded identical AO basis sets.

- RXCHF methods, modified version OCBSE procedure, only HF equation

- geminal-coupled electronic orbital projected orthogonal complement of the space spanned by the occupied regular electronic orbitals.
- This procedure iteration i assuming orbitals from the previous iteration.
- adopt different notation subsection clarity .

- define ψ (i−1) as the geminal-coupled electronic orbital obtained from the (i − 1)th iteration and -LCB- ψ a (i 1) 1≤ ≤ N α -RCB- regular alpha electronic orbitals obtained from the (i 1)th iteration.
- From the previous iteration, density matrices enable calculation matrices F e1 F α , defined AO basis appendices.
- solve Hartree-Fock-Roothaan equation for the regular alpha electronic orbitals in the AO basis,

- results in alpha electronic orbitals current iteration, -LCB- ψ a (i) 1≤ ≤ N ebf-RCB- , orbitals ordered first N α occupied remaining N ebf N α .
- By diagonalization procedure solve Eq. (44) , set alpha electronic orbitals .
- Note Hartree-FockRoothaan equation for the regular electronic orbitals any constraints depending geminal-coupled electronic orbital, i.e., no off-diagonal Lagrange multipliers equation.
- energy is minimized with respect to the electronic orbitals without enforcing orthogonality to the geminal-coupled electronic orbital.
- constraint applied only Hartree-Fock-Roothaan equation for the geminal-coupled electronic orbital.

- step transform Hartree-Fock-Roothaan equation for the geminal-coupled electronic orbital AO basis,

- basis (denoted B (i) ) space space spanned by the occupied regular elec-

- comprised electronic orbitals regular electrons current iteration.
- Transforming Eq. (45) before diagonalization provides solution geminal-coupled electronic orbital, ψ (i) , is orthogonal to all of the occupied regular electronic orbitals,

- Following formalism presented in Refs.
- 50 51 , projection B (i) eliminates - Lagrange multipliers Hartree-Fock-Roothaan equation for the geminal-coupled electronic orbital.
- this procedure minimizes energy with respect to variations geminal-coupled electronic orbital s constraint this orbital is orthogonal to all of the occupied regular electronic orbitals.
- Note orthogonalization schemes framework RXCHF methods.
- scheme some calculations provided supplementary material.
- 58

## III. DISCUSSION AND PRACTICAL CONSIDERATIONS

- evaluation of the geminal integrals appendices discussed detail papers.
- 24 , extension of the McMurchie-Davidson approach 52 evaluate three-, four -, five-particle geminal integrals.
- 53 More efficient integral schemes implemented tested, including resolution of identity (RI) approximations, Rys approach, 54 range tensor hypercontraction methods.
- 55 -- 57 These schemes expected offer significant advantages in computational tractability XCHF and RXCHF methods.

- choice GTG function parameters positronic systems discussed accompanying paper .
- 49 , demonstrate geminal parameters optimization one-electron-one-positron model system other positron-containing systems studied.
- expect transferability other types systems (e.g., proton-containing systems) nature electron-nucleus interaction, GTG parameters electron-proton interactions different electron-positron interactions.
- set GTG parameters type of quantum particle, parameters systems containing type of quantum particle.

- implementation RXCHF-fe limited N< 4 electrons and requires the evaluation four - particle integrals.
- discussed , computational expense XCHF calculations systems same size.
- RXCHF-fe possesses different underlying physical assumptions.
- XCHF method, all electronic orbitals are explicitly correlated to the nuclear orbitals same manner same GTG parameters.
- this case, geminal functions account interactions other than electron-nucleus dynamical correlation.
- RXCHF-fe method avoids problem by correlating only electrons to the quantum nuclei, ensuring geminal parameters short-ranged electron-nucleus interaction.
- XCHF wavefunction more optimized, RXCHF-fe wavefunction provides more description short-ranged electron-nucleus interaction.

- introduction RXCHF-ne and RXCHF-ae approximate methods provides more approaches based same underlying principles more RXCHF-fe method.
- These methods involve calculation only up to three-particle integrals avoid four - five-particle integrals required for the XCHF and RXCHF-fe methods.
- Combined more efficient integral techniques basis sets geminal-coupled electronic orbitals, RXCHF-ne and RXCHF-ae methods provide prospect studying larger chemical systems.
- accuracy tested smaller systems by comparison RXCHF-fe determine effects approximating electronic exchange interactions.
- Developing testing schemes approximate exchange interactions important direction research.

- Extensions frozen-core core potential methods directions.

- accompanying paper 49 demonstrates application of the RXCHF methods positron-containing molecular species, where the electrons and positrons are treated quantum mechanically .
- all RXCHF methods outperform XCHF and RXCHF-ne and RXCHF-ae methods approximations to the RXCHF-fe method.
- results provide outlook application implementations RXCHF methods to larger chemical systems.
- research directions focus extending formalism presented this paper case more than one geminalcoupled electronic orbital.
- ma jor goal develop RXCHF methods investigation proton-containing molecular species, where the electrons and select protons are treated quantum mechanically .

## IV . CONCLUSIONS

- this paper , presented RXCHF approach alternative developed XCHF approach including explicit electron-nucleus correlation within the NEO framework.
- paradigm shift restricting explicit electron-nucleus correlation only select electronic orbitals.
- RXCHF-fe, RXCHF-ne, RXCHF-ae methods defined compared terms degree electronic exchange contributions computational expense.
- Working expressions method derived reported form modified Fock operators sets Hartree-Fock-Roothaan equations.
- RXCHF-ne and RXCHF-ae methods offer substantial computational savings XCHF approach.
- Based success positron-containing molecular species, accompanying paper , 49 outlook application RXCHF methods to larger chemical systems .
- important direction application of the RXCHF methods chemical systems in which nuclear quantum effects , involving proton-coupled electron transfer reactions, exhibit non-Born-Oppenheimer effects between electrons and transferring proton(s).
- 4 , 5 cases, more than electronic orbital be explicitly correlated to the quantum nucleus.
- Extensions enable calculations are currently under development.

- This paper based work supported by National Science Foundation Grant No.
- CHE-10-57875 Air Force Office Scientific Research AFOSR Award No.
- FA9550-10-1-0081.
- A.S. thanks Natural Sciences Engineering Research Council Canada PGS scholarship.

## APPENDIX A: RXCHF-fe FOCK EXPRESSIONS

- this appendix, we report expressions for the Fock operators in the spin orbital, spatial orbital, atomic orbital bases for the RXCHF-fe method.
- operators defined

- appendices all expressed forms denoted by tilde.
- Details of the symmetrization procedure for all RXCHF methods given supplementary material.
- 58

## 1. Spin orbital basis

- quantum nuclear Fock operator is obtained by varying the energy in Eq. (23) respect p , leading

- variables ordered p , 1 , 2 , .
- g .
- ˜ (1) 2 ≡ ˜ (1) 2 ( p, 1, 2).

## 3. Atomic orbital basis

- We now expand the spatial orbitals in the AO bases, as in Eqs.
- (40) -- (43) .
- first define density matrices

- integrals required for the evaluation of these quantities are defined

## APPENDIX B: RXCHF-ne FOCK EXPRESSIONS

- this appendix, we report expressions for the Fock operators in the spin orbital, spatial orbital, atomic orbital bases for the RXCHF-ne method.

## 1. Spin orbital basis

- quantum nuclear Fock operator is obtained by varying the energy in Eq. (24) respect p , leading 1

- special electronic Fock operator is obtained by varying the energy in Eq. (24) respect χ 1 , leading e1 ( x 1 ) = S RXCHF 1 - ne ( χ p ˜ 1 χ p + =2 χ χ ˜ 2 χ χ

- + S RXCHF 1 - ne χ p χ 1 ∣ ∣ ̃ 2 ∣ ∣ χ p χ 1 , (B3) J K usual electronic Coulomb and exchange operators, , orbital χ .

## 2. Spatial orbital basis

- integrate RXCHF-ne Fock operators in Eqs.
- (B1) -- (B3) over spin to obtain the spatial Fock operators.
- RXCHF-ne allows possibility restricted Hartree-Fock (RHF) approach closed-shell treatment regular electrons, report open-shell expressions UHF approach RXCHF-fe expressions generality .

- case N α =N β , fα =fβ , modified Fock operators RHF formalism.

## 3. Atomic orbital basis

- We now expand the spatial orbitals in the AO bases, as in Eqs.
- (40) -- (43) .
- nuclear Fock operator in the AO basis is given by

- regular electronic Fock operator for α - spin orbitals in the AO basis is given by

- regular electronic Fock operator for β - spin orbitals in the AO basis is given by

- Most integrals required for the evaluation of these quantities are defined in the previous appendix, new integrals appearing are defined as

- quantum nuclear Fock operator is obtained by varying the energy in Eq. (30) respect p , leading 1

- − E S RXCHF G + E - ne χ 1 e ∣ ∣ g 2 ∣ ∣ χ 1 e .
- special electronic Fock operator is obtained by varying the energy in Eq. (30) respect χ 1 e , leading

- electronic Fock operator is obtained by varying the energy in Eq. (30) respect some χ μ ,

## 2. Spatial orbital basis

- integrate RXCHF-ae Fock operators in Eqs.
- (C1) -- (C3) over spin to obtain the spatial Fock operators.
- quantum nuclear Fock operator given by

## APPENDIX C: RXCHF-ae FOCK EXPRESSIONS

- this appendix, we report expressions for the Fock operators in the spin orbital, spatial orbital, atomic orbital bases for the RXCHF-ae method.

## 3. Atomic orbital basis

- We now expand the spatial orbitals in the AO bases, as in Eqs.
- (40) -- (43) .
- nuclear Fock operator in the AO basis is given by

- regular electronic Fock operator for α - spin orbitals in the AO basis is given by

- regular electronic Fock operator for β - spin orbitals in the AO basis is given by

- Most integrals required for the evaluation of these quantities are defined in the previous appendix, new integral appearing is defined as