4.9.4 Holomorphic Hartree–Fock Theory

To use multiple SCF solutions in NOCI (see Section 7.4), it is essential that all solutions exist across all geometries of interest to prevent discontinuities in the NOCI energies. However, it is well known that symmetry-broken SCF solutions can disappear at certain points along a potential energy surface, for example at the Coulson–Fischer point in H${}_2$. The holomorphic Hartree–Fock approach provides a means of analytically continuing solutions across all geometries.^{385, 129, 127}

In holomorphic Hartree–Fock theory, the real Hartree–Fock equations are analytically continued into the complex plane without introducing the complex conjugation of molecular orbital coefficients. Multiple solutions are then identified as the stationary points of the holomorphic energy³⁸⁵

$$\tilde{E}=\frac{⟨{\mathrm{\Psi }}^{*}|\widehat{H}|\mathrm{\Psi }⟩}{⟨{\mathrm{\Psi }}^{*}|\mathrm{\Psi }⟩},$$

(4.61)

where $\widehat{H}$ defines the conventional electronic Hamiltonian

(4.62)

As a result, the holomorphic Hartree–Fock equations are complex-analytic in the orbital coefficients, satisfying the Cauchy-Riemann conditions, and the number of stationary points is found to be constant across all geometries.¹²⁷ Real Hartree–Fock states remain stationary points of the holomorphic Hartree–Fock energy, and where real solutions vanish, their holomorphic counterparts continue to exist with complex orbital coefficients.^{129, 127}

Holomorphic Hartree–Fock stationary points can be located using minor modifications to conventional SCF algorithms.¹²⁹. Most significantly, by removing the complex conjugate of the wave function in Eq. (4.61), the required complex holomorphic density $𝐏$ and Fock $𝐅$ matrices become complex-symmetric (cf. Hermitian), satisfying $P^{\mu \nu }=P^{\nu \mu }$ and $F_{\mu \nu }=F_{\nu \mu }$. Moreover, since the complex conjugation must also be removed from the normalisation constraint, the molecular orbital coefficients must form a complex-orthogonal set (cf. unitary), i.e.

$$\sum _{\mu \nu }{C}_{i\cdot }^{\cdot \mu }{S}_{\mu \nu }{C}_{\cdot j}^{\nu \cdot }={\delta }_{ij},$$

(4.63)

Finally, the holomorphic Hartree–Fock orbital energies and total energy can in general also become complex, and thus selecting the new occupied orbitals on each SCF cycle using the orbital energies is poorly defined. Instead, a complex-symmetric analogue to the Maximum Overlap Method can be employed (see Section 4.5.6).

Following real solutions past the Coulson–Fischer point into the complex plane can often be difficult due to their coalesence with symmetry-pure solutions on the real axis. However, by scaling the electron-electron interaction using a complex parameter $\lambda$, i.e. introducing the Hamiltonian

(4.64)

it is possible to show that Coulson–Fischer points form isolated exceptional points on the real axis.¹²⁸ Consequently, following a suitable complex $\lambda$ trajectory allows real solutions to be perturbed off the real axis and followed with ease past the Coulson–Fischer point into their complex holomorphic regimes.^Burtonp:2019 These perturbed solutions can then be relaxed onto the real axis to identify the holomorphic Hartree–Fock states required for NOCI.

Within Q-Chem, the holomorphic Hartree–Fock approach is implemented in the LIBNOCI package (see Section 7.4.0.1), accessed using USE_LIBNOCI = TRUE and designed for locating multiple SCF solutions for use in NOCI calculations.