Analytic Nuclear Hessians

The fully analytical RHF Hessian includes five contributions:

$\frac{\partial ^{2} E}{\partial X _{A} \partial X _{B}} = H^{(1 e)} + H^{(S)} + H^{(ERI)} + H^{(CPHF)} + H^{(oo)}$

Contributions

Term	Formula	Description
$H^{(1 e)}$	$\frac{\partial ^{2} H _{core}}{\partial X _{A} \partial X _{B}}$	One-electron curvature: second derivatives of $H_{core}$
$H^{(S)}$	$\frac{\partial ^{2} S}{\partial X _{A} \partial X _{B}}$ contracted with $W$	Overlap curvature
$H^{(ERI)}$	$\frac{\partial ^{2} ( μν ∣ λσ )}{\partial X _{A} \partial X _{B}}$	ERI curvature: second derivatives of two-electron integrals computed analytically via `shell_quartet_eri_hessian` (not finite-difference)
$H^{(CPHF)}$	$2 \cdot Tr [F^{(x)} \cdot U^{(y)}] + 2 \cdot Tr [F^{(y)} \cdot U^{(x)}]$	CPHF response: $U$ is the orbital response from iterative CPHF solution
$H^{(oo)}$	$Tr [P_{oo} F_{skel}] - Tr [W_{oo} S]$	Occupied-occupied reorthonormalization: includes both $d C / d y$ and $d ε / d y$ contributions to $W_{oo}$

UHF Hessian

For UHF, the Hessian uses a coupled 2×2 CPHF system ( $αα$ , $α β$ , $β α$ , $ββ$ spin blocks) with separate $α$ and $β$ densities.

The semi-analytical (finite-difference of gradients) path propagates the selected ERI engine to all displaced SCF evaluations. This ensures consistency between the analytical and numerical Hessian paths.

Symmetry Acceleration

Analytical Hessians skip symmetry-equivalent shell quartets before expensive integral evaluation, matching GRAD2E SymShl. This provides significant speedup for symmetric molecules.

Usage

FREQ
  Hessian  analytical  # or semi
  ...
END

The analytical Hessian is the default for RHF/UHF single-point frequency calculations.

OpenQuantum

Analytic Nuclear Hessians

Contributions

UHF Hessian

Semi-Analytical Path

Symmetry Acceleration

Usage