4.2 Theoretical Background 4.2.1 SCF and LCAO Approximations 4.3 Basic SCF Job Control

4.2.2 Hartree-Fock Theory

(September 1, 2024)

As with much of the theory underlying modern quantum chemistry, the HF approximation was developed shortly after publication of the Schrödinger equation, but remained a qualitative theory until the advent of the computer. Although the HF approximation tends to yield qualitative chemical accuracy, rather than quantitative information, and is generally inferior to many of the DFT approaches available, it remains as a useful tool in the quantum chemist’s toolkit. In particular, for organic chemistry, HF predictions of molecular structure are very useful.

Consider once more the Roothaan-Hall equations, Eq. (4.11), or the Pople-Nesbet equations, Eq. (4.13), which can be traced back to Eq. (4.8), in which the effective potential $\upsilon_{\rm eff}$ depends on the SCF methodology. In a restricted HF (RHF) formalism, the effective potential can be written as

\upsilon_{\rm eff}=\sum_{a}^{N/2}\bigl{[}{2\hat{J}_{a}(1)-\hat{K}_{a}(1)}\bigr% {]}-\sum_{A=1}^{M}{\frac{Z_{A}}{r_{1A}}}

(4.15)

where the Coulomb and exchange operators are defined as

\hat{J}_{a}(1)=\int{\psi_{a}^{\ast}(2)\frac{1}{r_{12}}\psi_{a}(2)\;d{\rm{\bf r% }}_{2}}

(4.16)

and

\hat{K}_{a}(1)\psi_{i}(1)=\left[{\int{\psi_{a}^{\ast}(2)\frac{1}{r_{12}}\psi_{% i}(2)\;d{\rm{\bf r}}_{2}}}\right]\psi_{a}(1)

(4.17)

respectively. By introducing an atomic orbital basis, we obtain Fock matrix elements

F_{\mu\nu}=H_{\mu\nu}^{\mathrm{core}}+J_{\mu\nu}-K_{\mu\nu}

(4.18)

where the core Hamiltonian matrix elements

H_{\mu\nu}^{\mathrm{core}}=T_{\mu\nu}+V_{\mu\nu}

(4.19)

consist of kinetic energy elements

T_{\mu\nu}=\int{\phi_{\mu}({\rm{\bf r}})\left(-\frac{1}{2}\hat{\nabla}^{2}% \right)\phi_{\nu}({\rm{\bf r}})\;d{\rm{\bf r}}}

(4.20)

and nuclear attraction elements

V_{\mu\nu}=\int{\phi_{\mu}({\rm{\bf r}})\left(-\sum_{A}\frac{Z_{A}}{\left|{{% \rm{\bf R}}_{A}-{\rm{\bf r}}}\right|}\right)\phi_{\nu}({\rm{\bf r}})\;d{\rm{% \bf r}}}

(4.21)

The Coulomb and exchange elements are given by

J_{\mu\nu}=\sum\limits_{\lambda\sigma}{P_{\lambda\sigma}\left({\mu\nu|\lambda% \sigma}\right)}

(4.22)

and

K_{\mu\nu}=\frac{1}{2}\sum\limits_{\lambda\sigma}{P_{\lambda\sigma}\left({\mu% \lambda|\nu\sigma}\right)}

(4.23)

respectively, where the density matrix elements are

P_{\mu\nu}=2\sum\limits_{a=1}^{N/2}{C_{\mu a}C_{\nu a}}

(4.24)

and the two electron integrals are

\left({\mu\nu|\lambda\sigma}\right)=\int\int{\phi_{\mu}({\rm{\bf r}}_{1})\phi_% {\nu}({\rm{\bf r}}_{1})\left({\frac{1}{r_{12}}}\right)\phi_{\lambda}({\rm{\bf r% }}_{2})\phi_{\sigma}({\rm{\bf r}}_{\rm{\bf 2}})\;d{\rm{\bf r}}_{1}\;d{\rm{\bf r% }}_{2}}\;.

(4.25)

Note: The formation and utilization of two-electron integrals is a topic central to the overall performance of SCF methodologies. The performance of the SCF methods in new quantum chemistry software programs can be quickly estimated simply by considering the quality of their atomic orbital integrals packages. See Appendix A for details of Q-Chem’s AOInts package.

Substituting the matrix element in Eq. (4.18) back into the Roothaan-Hall equations, Eq. (4.11), and iterating until self-consistency is achieved will yield the RHF energy and wave function. Alternatively, one could have adopted the unrestricted form of the wave function by defining separate $\alpha$ and $\beta$ density matrices:

\displaystyle\begin{aligned} \displaystyle P_{\mu\nu}^{\alpha}&\displaystyle=% \sum\limits_{a=1}^{n_{\alpha}}{C_{\mu a}^{\alpha}C_{\nu a}^{\alpha}}\\ \displaystyle P_{\mu\nu}^{\beta}&\displaystyle=\sum\limits_{a=1}^{n_{\beta}}{C% _{\mu a}^{\beta}C_{\nu a}^{\beta}}\end{aligned}

(4.26)

The total electron density matrix $\mathbf{P}=\mathbf{P}^{\alpha}+\mathbf{P}^{\beta}$ . The unrestricted $\alpha$ Fock matrix,

F_{\mu\nu}^{\alpha}=H_{\mu\nu}^{\mathrm{core}}+J_{\mu\nu}-K_{\mu\nu}^{\alpha}\;,

(4.27)

differs from the restricted one only in the exchange contributions, where the $\alpha$ exchange matrix elements are given by

K_{\mu\nu}^{\alpha}=\sum\limits_{\lambda}^{N}{\sum\limits_{\sigma}^{N}{P_{% \lambda\sigma}^{\alpha}\left({\mu\lambda|\nu\sigma}\right)}}

(4.28)

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4.2.2 Hartree-Fock Theory