Ab initio potential energy surfaces calculation via restricted Hartree–Fock for molecular dynamics simulation: a comprehensive review

Abstract

This study systematically analyzes the restricted Hartree–Fock (RHF) method, which is used to calculate the potential energy surface (PES) of closed-shell molecular systems. In this process, an optimized algorithm for the calculation of the RHF matrix based on the Roothaan equation is designed and a detailed programming strategy for its implementation is presented. The algorithm is highly stable in the Psi4 library and Python environment and demonstrates the ability to efficiently calculate PES for basic organic compounds such as ethane (C₂H₆), propane (C₃H₈), and butane (C₄H₁₀). Furthermore, the PES derived by the proposed algorithm plays a crucial role in determining the force field parameters for CHARMM, AMBER, COMPASS, and DREIDING, which are essential for molecular dynamics simulations. The findings of this study provide fundamental insights into the study of first-principle-based molecular dynamics simulations, which are expected to expand the scope of applications in basic research for an ab initio computational chemistry and physics.

Appendices

Appendix A. derivation Eq. (2.1c)

$$ \begin{gathered} E_{elec} = \left\langle {{\Psi }_{0} \left| {{\hat{\text{H}}}} \right.\left| {{\Psi }_{0} } \right.} \right\rangle = \left\langle {{\Psi }_{0} \left| {{\text{h}}\left( 1 \right) + {\text{h}}\left( 2 \right) + \frac{1}{{{\text{r}}_{12} }}\left| {{\Psi }_{0} } \right.} \right.} \right\rangle \hfill \\ = \underbrace {{\left\langle {{\Psi }_{0} \left| {{\text{h}}\left( 1 \right)} \right.\left| {{\Psi }_{0} } \right.} \right\rangle }}_{{\text{a}}} + \underbrace {{\left\langle {{\Psi }_{0} \left| {{\text{h}}\left( 2 \right)\left| {{\Psi }_{0} } \right.} \right.} \right\rangle }}_{{\text{b}}} + \underbrace {{\left\langle {{\Psi }_{0} \left| {\frac{1}{{{\text{r}}_{12} }}\left| {{\Psi }_{0} } \right.} \right.} \right\rangle }}_{{\text{c}}} \hfill \\ \end{gathered} $$

(A.1)

By substituting Eq. (2.1b) with the term a in Eq. (A.1) and expanding it, the following equation can be obtained

$$\langle {\Psi }_{0}|h(1)|{\Psi }_{0}\rangle =\int {dx}_{1}{dx}_{2}\frac{1}{\sqrt{2}}{\left\{{X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{b}\left(1\right){X}_{a}\left(2\right)\right\}}^{*}h\left(1\right)\frac{1}{\sqrt{2}}\left\{{X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{b}\left(1\right){X}_{a}\left(2\right)\right\} =\frac{1}{2}\int {dx}_{1}{dx}_{2}\left\{{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right)h\left(1\right){X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right)h\left(1\right){X}_{b}\left(1\right){X}_{a}\left(2\right) -{X}_{b}^{*}\left(1\right){X}_{a}^{*}\left(2\right)h\left(1\right){X}_{a}\left(1\right){X}_{b}\left(2\right)+{X}_{b}^{*}\left(1\right){X}_{a}^{*}\left(2\right)h\left(1\right){X}_{b}\left(1\right){X}_{a}\left(2\right)\right\} $$

(A.2)

When the orthonormalization conditions, $\int {dx}_{2}{X}_{b}\left(2\right){X}_{b}\left(2\right)=1$ and $\int {dx}_{2}{X}_{b}\left(2\right){X}_{a}\left(2\right)=0$ of the wavefunction are applied to eliminate ${dx}_{2}$ on the right-hand side, the energy expectation value for $h(1)$, i.e., the term a in Eq. (A.1), can be calculated as

$$\langle {\Psi }_{0}|h(1)|{\Psi }_{0}\rangle =\frac{1}{2}\int {dx}_{1}{X}_{a}^{*}\left(1\right)h(1){X}_{a}(1)+\frac{1}{2}\int {dx}_{1}{X}_{b}^{*}h\left(1\right){X}_{b}\left(1\right)$$

(A.3)

The term b in Eq. (A.1) can be calculated in the same way as the term a, and the result is the same as shown in Eq. (A.3). Thus, $h\left(1\right)+h(2)$, which corresponds to the 1e⁻ operator in Eq. (A.1), can finally be generalized as

$$\langle {\Psi }_{0}|h\left(1\right)+h(2)|{\Psi }_{0}\rangle =\int dx{ X}_{a}^{*} h {X}_{a}+\int dx {X}_{b}^{*} h {X}_{b}\to \sum_{a}^{N}\langle {X}_{a}|h|{X}_{a}\rangle $$

(A.4)

Next, by substituting Eq. (2.1b) with the term c in Eq. (A.1), the following equation can be obtained.

$$\langle {\Psi }_{0}|\frac{1}{{r}_{12}}|{\Psi }_{0}\rangle =\int {dx}_{1}{dx}_{2}\frac{1}{\sqrt{2}}{\left\{{X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{b}\left(1\right){X}_{a}\left(2\right)\right\}}^{*}\frac{1}{{r}_{12}}\frac{1}{\sqrt{2}}\left\{{X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{b}\left(1\right){X}_{a}\left(2\right)\right\}==\frac{1}{2}\int {dx}_{1}{dx}_{2}\left\{{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right){\frac{1}{{r}_{12}}X}_{a}\left(1\right){X}_{b}\left(2\right)-{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right)\frac{1}{{r}_{12}}{X}_{b}\left(1\right){X}_{a}\left(2\right) -{X}_{b}^{*}\left(1\right){X}_{a}^{*}\left(2\right)\frac{1}{{r}_{12}}{X}_{a}\left(1\right){X}_{b}\left(2\right)+{X}_{b}^{*}\left(1\right){X}_{a}^{*}\left(2\right)\frac{1}{{r}_{12}}{X}_{b}\left(1\right){X}_{a}\left(2\right)\right\}$$

(A.5)

Since (1) and (2) can be changed with dummy indices, Eq. (A.6) can be obtained by exchanging the electron indices in the third and fourth terms of Eq. (A.5).

$$\langle {\Psi }_{0}|\frac{1}{{r}_{12}}|{\Psi }_{0}\rangle =\int {dx}_{1}{dx}_{2}{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right){\frac{1}{{r}_{12}}X}_{a}\left(1\right){X}_{b}\left(2\right)-\int {dx}_{1}{dx}_{2}{X}_{a}^{*}\left(1\right){X}_{b}^{*}\left(2\right){\frac{1}{{r}_{12}}X}_{b}\left(1\right){X}_{a}\left(2\right)$$

(A.6)

$$ \to \left\langle {X_{a} X_{b} {|}X_{a} X_{b} } \right\rangle - \left\langle {X_{a} X_{b} {|}X_{b} X_{a} } \right\rangle $$

(A.7)

$$ \to \frac{1}{2}\mathop \sum \limits_{a}^{N} \mathop \sum \limits_{b}^{N} \left\langle {X_{a} X_{b} {|}X_{a} X_{b} } \right\rangle - \left\langle {X_{a} X_{b} {|}X_{b} X_{a} { }} \right\rangle { } $$

(A.8)

By Dirac notation, Eq. (A.7) can be written as a representation of Eq. (A.6), while Eq. (A.8) is a generalization of Eq. (A.7) for N electrons. On multiplying Eq. (A.8) by ½, the effect of double counting can be eliminated since a- > b and b- > a are the same interactions. From the above expansion, E_elec based on spin-orbitals can be finally expressed by adding Eqs. (A.4) and (A.8) as

$${E}_{elec}=\sum_{a}^{N}\langle {X}_{a}|h|{X}_{a}\rangle +\frac{1}{2}\sum_{a}^{N}\sum_{b}^{N}\langle {X}_{a}{X}_{b}|{X}_{a}{X}_{b}\rangle -\langle {X}_{a}{X}_{b}|{X}_{b}{X}_{a}\rangle $$

(A.9)

Appendix B. derivation Eq. (3.1e)

$$\widehat{f}\left({r}_{1}\right)=\int {dw}_{1}{\alpha }^{*}\left({w}_{1}\right)\left\{h\left({r}_{1}\right)+\sum_{b}^{N}\int {dx}_{2}{X}_{b}^{*}\left(2\right)\frac{1}{{r}_{12}}\left(1-{P}_{12}\right){X}_{b}\left(2\right)\right\}\alpha \left({w}_{1}\right)$$

$$=\int {dw}_{1}{\alpha }^{*}\left({w}_{1}\right)h\left({r}_{1}\right)\alpha \left({w}_{1}\right)+\sum_{b}^{N}\int {dw}_{1}{dx}_{2}{\alpha }^{*}\left({w}_{1}\right){X}_{b}^{*}\left({x}_{2}\right)\frac{1}{{r}_{12}}{X}_{b}\left({x}_{2}\right)\alpha \left({w}_{1}\right)$$

$$-\sum_{b}^{N}\int {dw}_{1}{dx}_{2}{\alpha }^{*}\left({w}_{1}\right){X}_{b}^{*}\left({x}_{2}\right)\frac{1}{{r}_{12}}{X}_{b}\left({x}_{1}\right)\alpha \left({w}_{2}\right)$$

(B.1)

By applying the orthonormalization condition, $\int {dw}_{1}{\alpha }^{*}\left({w}_{1}\right)\alpha \left({w}_{1}\right)=1$, for the spin function in Eq. (B.1), and further substituting $\sum_{b}^{N}{X}_{b}$ with Eq. (3.0a) to convert the spin-orbitals to space-orbitals, the following equation can be obtained.

$$\widehat{f}\left({r}_{1}\right)=h\left({r}_{1}\right)+\sum_{b}^{N/2}\int {dw}_{1}{dw}_{2}{dr}_{2}{\alpha }^{*}({w}_{1}){\alpha }^{*}({w}_{2}){\psi }^{*}({r}_{2})\frac{1}{{r}_{12}}\psi ({r}_{2})\alpha ({w}_{2})\alpha ({w}_{1})+\sum_{b}^{N/2}\int {dw}_{1}{dw}_{2}{dr}_{2}{\alpha }^{*}({w}_{1}){\beta }^{*}({w}_{2}){\psi }^{*}({r}_{2})\frac{1}{{r}_{12}}\psi ({r}_{2})\beta ({w}_{2})\alpha ({w}_{1})-\sum_{b}^{N/2}\int {dw}_{1}{dw}_{2}{dr}_{2}{\alpha }^{*}({w}_{1}){\alpha }^{*}({w}_{2}){\psi }^{*}({r}_{2})\frac{1}{{r}_{12}}\psi ({r}_{1})\alpha ({w}_{1})\alpha ({w}_{2})-\sum_{b}^{N/2}\int {dw}_{1}{dw}_{2}{dr}_{2}{\alpha }^{*}({w}_{1}){\beta }^{*}({w}_{2}){\psi }^{*}({r}_{2})\frac{1}{{r}_{12}}\psi ({r}_{1})\beta ({w}_{1})\alpha ({w}_{2})$$

(B.2)

When the orthonormalization condition is applied to the spin function again in Eq. (B.3), the first, second, and third sigma terms are eliminated by the condition that the spin function becomes 1, leaving only the space-function. Furthermore, the last term is eliminated by the conditions of $\int {dw}_{1}{\alpha }^{*}\left({w}_{1}\right)\alpha \left({w}_{2}\right)=0$ and $\int {dw}_{1}{\beta }^{*}\left({w}_{1}\right)\beta \left({w}_{2}\right)=0$. Thus, the closed-shell Fock operator $\widehat{f}({r}_{1})$ can be expressed as

$$\widehat{f}\left({r}_{1}\right)=h\left({r}_{1}\right)+\sum_{b}^{N/2}2\int {dr}_{2}{\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}\psi \left({r}_{2}\right)-\sum_{b}^{N/2}\int {dr}_{2}{\psi }^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}\psi ({r}_{1})$$

(B.3)

If the permutation operator (${P}_{12}$) is applied to the third term in the above equation, $\widehat{f}({r}_{1})$ can be simplified as

$$\widehat{f}\left({r}_{1}\right)=h\left({r}_{1}\right)+\sum_{b}^{N/2}\int {dr}_{2}{\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}(2-{P}_{12})\psi \left({r}_{2}\right) \to h\left({r}_{1}\right)+\sum_{b}^{N/2}2{J}_{b}\left({r}_{1}\right)-{K}_{b}({r}_{1})$$

(B.4)

Appendix C. derivation Eq. (3.3c)

By substituting Eq. (3.3b) into Eq. (3.2c), the following equation can be obtained.

$$FX{C}{\prime}=SXC{\prime}\varepsilon $$

(C.1)

If both sides are multiplied by the conjugate transpose of X, Eq. (C.1) can be expanded as

$$ X^{\dag } FXC^{\prime} = X^{\dag } SXC^{\prime}\varepsilon $$

(C.2)

$$ = \left( {Us^{ - 1/2} } \right)^{\dag } S\left( {Us^{ - 1/2} } \right)C^{\prime}\varepsilon $$

(C.3)

$$ = s^{ - 1/2} U^{\dag } SUs^{ - 1/2} C^{\prime}\varepsilon $$

(C.4)

In Eq. (C.4), $U^{\dag } SU$ is the diagonal matrix s of S. Thus, $s^{ - 1/2} U^{\dag } SUs^{ - 1/2}$ is equal to 1. If we define $X^{\dag } FX\,as\,F^{\prime}$, the left-hand side of Eq. (C.2), then the Roothaan equation is finally expressed as

$$F{\prime}{C}{\prime}=C{\prime}\varepsilon $$

(C.5)

Appendix D. derivation Eq. (3.4c)

$${G}_{\mu \nu }=\sum_{b}^{N/2}\left\{2\int {dr}_{1}{dr}_{2}{\phi }_{\upmu }^{*}({r}_{1}){\psi }_{b}^{*}({r}_{2})\frac{1}{{r}_{12}}{{\phi }_{\nu }({r}_{1})\psi }_{b}({r}_{2})-\int {dr}_{1}{dr}_{2}{\phi }_{\upmu }^{*}({r}_{1}){\psi }_{b}^{*}({r}_{2})\frac{1}{{r}_{12}}{{\psi }_{b}({r}_{1})\phi }_{\nu }({r}_{2})\right\}$$

(D.1)

Here, the electrons in orbit b are still represented by space-wave function. Thus, if we apply a linear combination of the Gaussian basis functions of Eq. (3.2a) to convert them into a form that allows for matrix operations, ${\psi }_{b}^{*}$ and ${\psi }_{b}$ can be represented by ${\psi }_{b}^{*}\left({r}_{2}\right)=\sum_{\lambda }^{K}{C}_{\lambda b}^{*}{\phi }_{\lambda }$ and ${\psi }_{b}\left({r}_{2}\right)={\sum }_{\sigma }^{K}{C}_{\sigma b}{\phi }_{\sigma }$, respectively. By substituting them into Eq. (D.1) and expanding it, the following equation can be obtained.

$${G}_{\mu \upsilon }=\sum_{b}^{N/2}\sum_{\sigma \lambda }^{K}\left[2{C}_{\lambda b}^{*}{C}_{\sigma b}\int {dr}_{1}{dr}_{2}{\phi }_{\upmu }^{*}\left({r}_{1}\right){\phi }_{\lambda }^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\phi }_{\nu }\left({r}_{1}\right){\phi }_{\sigma }\left({r}_{2}\right)-{C}_{\lambda b}^{*}{C}_{\sigma b}\int {dr}_{1}{dr}_{2}{\phi }_{\upmu }^{*}\left({r}_{1}\right){\phi }_{\lambda }^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\phi }_{\sigma }\left({r}_{1}\right){\phi }_{\nu }\left({r}_{2}\right)\right] =\sum_{b}^{N/2}\sum_{\sigma \lambda }^{K}{C}_{\lambda b}^{*}{C}_{\sigma b}\left[2\langle {\phi }_{\upmu }{\phi }_{\lambda }|{\phi }_{\nu }{\phi }_{\sigma }\rangle -\langle {\phi }_{\upmu }{\phi }_{\lambda }|{\phi }_{\sigma }{\phi }_{\nu }\rangle \right] $$

(D.2)

By applying the electron density matrix from Eq. (3.4d) to Eq. (D.2) and summarizing it, term ${G}_{\mu \nu }$ can be simplified as

$${G}_{\mu \nu }=\sum_{\sigma \lambda }^{K}{P}_{\lambda \sigma }\left[\langle {\phi }_{\mu }{\phi }_{\lambda }|{\phi }_{\nu }{\phi }_{\sigma }\rangle -\frac{1}{2}\langle {\phi }_{\mu }{\phi }_{\lambda }|{\phi }_{\sigma }{\phi }_{\nu }\rangle \right]$$

(D.3)

Appendix E. derivation Eq. (3.5a)

$${E}_{elec}=\sum_{a}^{N}\langle {X}_{a}|h|{X}_{b}\rangle +\sum_{a}^{N}\sum_{b}^{N}\langle {X}_{a}{X}_{b}|{X}_{a}{X}_{b}\rangle -\langle {X}_{a}{X}_{b}|{X}_{b}{X}_{a}\rangle $$

(E.1)

By substituting Eq. (3.0a) for the 1e⁻ operator, the first term in Eq. (E.1) can be converted to a function of the spatial-orbital as

$$\sum_{a}^{N}\langle {X}_{a}|h|{X}_{b}\rangle =\sum_{a}^{N/2}\langle {\psi }_{a}\alpha |h|{\psi }_{a}\alpha \rangle +\sum_{a}^{N/2}\langle {\psi }_{a}\beta |h|{\psi }_{a}\beta \rangle $$

$$=\sum_{a}^{N/2}\int {dr}_{1}{dw}_{1}{\psi }_{a}^{*}\left({r}_{1}\right){\alpha }^{*}({w}_{1})h{\psi }_{a}({r}_{1})\alpha ({w}_{1})+\sum_{a}^{N/2}\int {dr}_{1}{dw}_{1}{\psi }_{a}^{*}\left({r}_{1}\right){\beta }^{*}\left({w}_{1}\right)h{\psi }_{a}\left({r}_{1}\right)\beta \left({w}_{1}\right)$$

(E.2)

By applying the orthonormalization condition $\int {dw}_{1}{\alpha }^{*}\left({w}_{1}\right)\alpha \left({w}_{1}\right)=1$ and $\int {dw}_{1}{\beta }^{*}\left({w}_{1}\right)\beta \left({w}_{1}\right)=1$ for the spin function in Eq. (E.2), the following equation can be obtained.

$$\sum_{a}^{N}\langle {X}_{a}|h|{X}_{b}\rangle =\sum_{a}^{N/2}\int {dr}_{1}{\psi }_{a}^{*}\left({r}_{1}\right)h{\psi }_{a}\left({r}_{1}\right)+\sum_{a}^{N/2}\int {dr}_{1}{dw}_{1}{\psi }_{a}^{*}\left({r}_{1}\right)h{\psi }_{a}\left({r}_{1}\right)$$

$$=\sum_{a}^{N/2}2\langle {\psi }_{a}|h|{\psi }_{a}\rangle $$

(E.3)

Next, by substituting Eq. (3.0b) for the 2e⁻ operator in Eq. (E.1), the second term can be converted into a function of the spatial-orbital as

$$\frac{1}{2}\sum_{a}^{N}\sum_{b}^{N}\langle {X}_{a}{X}_{b}|{X}_{a}{X}_{b}\rangle -\langle {X}_{a}{X}_{b}|{X}_{b}{X}_{a}\rangle $$

$$ \begin{gathered} \frac{1}{2}\{ \sum\limits_{a}^{1\backslash 2} {\sum\limits_{b}^{1\backslash 2} {\underbrace {{\left\langle {\psi_{a} \alpha { }\psi_{b} \alpha \left| {\psi_{a} \alpha { }\psi_{b} \alpha } \right.} \right\rangle }}_{1} - \underbrace {{\left\langle {\psi_{a} \alpha { }\psi_{b} \alpha \left| {\psi_{b} \alpha { }\psi_{a} \alpha } \right.} \right\rangle }}_{2}} } \hfill \\ + \sum\limits_{a}^{1\backslash 2} {\sum\limits_{b}^{1\backslash 2} {\underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \alpha \left| {\psi_{a} \beta { }\psi_{b} \alpha } \right.} \right\rangle }}_{1} - \underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \alpha \left| {\psi_{b} \beta { }\psi_{a} \beta } \right.} \right\rangle }}_{3}} } \hfill \\ + \sum\limits_{a}^{1\backslash 2} {\sum\limits_{b}^{1\backslash 2} {\underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \alpha \left| {\psi_{a} \beta { }\psi_{b} \alpha } \right.} \right\rangle }}_{1} - \underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \alpha \left| {\psi_{b} \beta { }\psi_{a} \beta } \right.} \right\rangle }}_{3}} } \hfill \\ + \sum\limits_{a}^{1\backslash 2} {\sum\limits_{b}^{1\backslash 2} {\underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \beta \left| {\psi_{a} \beta { }\psi_{b} \beta } \right.} \right\rangle }}_{1} - \underbrace {{\left\langle {\psi_{a} \beta { }\psi_{b} \beta \left| {\psi_{b} \beta { }\psi_{a} \beta } \right.} \right\rangle }}_{2}} } \} \hfill \\ \end{gathered} $$

(E.4)

Here, the orthonormalization condition of the spin function causes the terms labeled 1 in Eq. (E.4) to be $\left\langle {\psi_{a} \psi_{b} \left| {\psi_{a} \psi_{b} } \right.} \right\rangle$, the terms labeled 2 to be $\left\langle {\psi_{a} \psi_{b} \left| {\psi_{b} \psi_{a} } \right.} \right\rangle$, and the terms labeled 3 to be zero. Thus, the 2e⁻ operator in Eq. (E.1) can be expressed as

$$\frac{1}{2}\sum_{a}^{N}\sum_{b}^{N}\langle {X}_{a}{X}_{b}|{X}_{a}{X}_{b}\rangle -\langle {X}_{a}{X}_{b}|{X}_{b}{X}_{a}\rangle =\frac{1}{2}\left\{\sum_{a}^{1/2}\sum_{b}^{1/2}4\langle {\psi }_{a}{\psi }_{b}|{\psi }_{a}{\psi }_{b}\rangle -2\langle {\psi }_{a}{\psi }_{b}|{\psi }_{b}{\psi }_{a}\rangle \right\}=\sum_{ab}^{N/2}2\langle {\psi }_{a}{\psi }_{b}|{\psi }_{a}{\psi }_{b}\rangle -\langle {\psi }_{a}{\psi }_{b}|{\psi }_{b}{\psi }_{a}\rangle $$

(E.5)

Thus, E_elec, the total energy of electrons, based on the space-orbitals in a closed-shell system, is finally expressed using Eqs. (E.3) and (E.5) as

$${E}_{elec}=\sum_{a}^{N/2}2\langle {\psi }_{a}|h|{\psi }_{a}\rangle +\sum_{ab}^{N/2}2\langle {\psi }_{a}{\psi }_{b}|{\psi }_{a}{\psi }_{b}\rangle -\langle {\psi }_{a}{\psi }_{b}|{\psi }_{b}{\psi }_{a}\rangle $$

(E.6)

Appendix F. derivation Eq. (3.5b)

$${E}_{elec}=\sum_{a}^{N/2}2\langle {\psi }_{a}|h|{\psi }_{a}\rangle +\sum_{ab}^{N/2}2\langle {\psi }_{a}{\psi }_{b}|{\psi }_{a}{\psi }_{b}\rangle -\langle {\psi }_{a}{\psi }_{b}|{\psi }_{b}{\psi }_{a}\rangle =\sum_{a}^{N/2}\left(2\langle {\psi }_{a}|h|{\psi }_{a}\rangle +\sum_{b}^{N/2}2{J}_{ab}-{K}_{ab}\right)$$

(F.1)

Here, by substituting Eq. (3.2a) into the spatial-orbit function and expanding it, the following equation that allows for matrix operations can be obtained.

$${E}_{elec}=\sum_{a}^{N/2}\left(2\sum_{\mu \nu }^{K}{C}_{\mu a}^{*}{C}_{\nu a}\int {dr}_{1}{\phi }_{\mu }^{*}\left({r}_{1}\right)h{\phi }_{\nu }({r}_{1})+\sum_{b}^{N/2}\left(2\sum_{\mu \nu }^{K}{C}_{\mu a}^{*}{C}_{\nu a}\int {dr}_{1}{dr}_{2}{\phi }_{\mu }^{*}\left({r}_{1}\right){\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\phi }_{\nu }\left({r}_{1}\right){\psi }_{b}\left({r}_{2}\right)-\sum_{\mu \nu }^{K}{C}_{\mu a}^{*}{C}_{\nu a}\int {dr}_{1}{dr}_{2}{\phi }_{\mu }^{*}\left({r}_{1}\right){\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\psi }_{b}\left({r}_{1}\right){\phi }_{\nu }\left({r}_{2}\right)\right)\right)$$

$$=\sum_{a}^{N/2}\sum_{\mu \nu }^{K}{C}_{\mu a}^{*}{C}_{\nu a}\left(2\int {dr}_{1}{\phi }_{\mu }^{*}\left({r}_{1}\right)h{\phi }_{\nu }({r}_{1})+\sum_{b}^{N/2}\left(2\int {dr}_{1}{dr}_{2}{\phi }_{\mu }^{*}\left({r}_{1}\right){\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\phi }_{\nu }\left({r}_{1}\right){\psi }_{b}\left({r}_{2}\right)-\int {dr}_{1}{dr}_{2}{\phi }_{\mu }^{*}\left({r}_{1}\right){\psi }_{b}^{*}\left({r}_{2}\right)\frac{1}{{r}_{12}}{\psi }_{b}\left({r}_{1}\right){\phi }_{\nu }\left({r}_{2}\right)\right)\right)$$

(F.2)

Here, the first term can be expressed as ${H}_{\mu \nu }^{core}$ and the second term can be expressed as ${G}_{\mu \nu }$ by Eq. (3.4a). When we apply this and combine the common ${C}_{\mu a}^{*}{C}_{\nu a}$, Eq. (F.2) can be written as

$${E}_{elec}=\sum_{a}^{N/2}\sum_{\mu \nu }^{K}{C}_{\mu a}^{*}{C}_{\nu a}\left(2{H}_{\mu \nu }^{core}+{G}_{\mu \nu }\right)$$

(F.3)

Here, if we consider that ${F}_{\mu \nu }=$ ${H}_{\mu \nu }^{core}$ +${G}_{\mu \nu }$ (see Sect. 3.4) can be established, and apply the electron density matrix of Eq. (3.4d), E_elec calculated from the Fock operator can be finally expressed as

$${E}_{elec}=\frac{1}{2}\sum_{\mu \nu }^{K}{P}_{\mu \nu }\left[{H}_{\mu \nu }^{core}+{F}_{\mu \nu }\right] $$

(F.4)