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

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

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

Tables