- Generated by
1.16.1
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votca 2026-dev
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#include <dftgradient.h>
Static Public Member Functions | |
| static Eigen::MatrixXd | NuclearRepulsionDerivative (const QMMolecule &mol) |
| static Eigen::MatrixXd | RIJGradient (const Eigen::MatrixXd &density, const AOBasis &auxbasis, const AOBasis &dftbasis) |
| static Eigen::MatrixXd | RIKGradient (const Eigen::MatrixXd &occ_mo_coeffs, const AOBasis &auxbasis, const AOBasis &dftbasis) |
Definition at line 52 of file dftgradient.h.
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Derivative of the classical nuclear-nuclear repulsion energy E_nn = sum over atom pairs A,B with A < B of Z_A Z_B / R_AB, with respect to nuclear coordinates. Returns an (Natoms x 3) matrix in Hartree/Bohr, consistent with QMAtom::getPos() being stored in Bohr and with the Orbitals::setForces() convention – no unit conversion needed here, unlike the libint2-derived integral derivatives (which needed a Bohr/Angstrom fix in their own finite-difference tests): this function consumes QMAtom positions directly, already in Bohr, and produces a result already in atomic units end to end.
Uses QMAtom::getNuccharge() (nuclear charge minus ECP core charge, if an ECP is in use), the effective charge that should enter the classical repulsion term consistently with an ECP-modified Hamiltonian.
Definition at line 41 of file dftgradient.cc.
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RI-J (Coulomb) gradient assembly, contracting the already-validated two-/three-center integral derivatives with a FIXED density matrix and the corresponding RI fitting coefficients. Returns (Natoms x 3) dE_J/dR in Hartree/Bohr, same convention as NuclearRepulsionDerivative.
IMPORTANT: "density" here does not need to come from a converged SCF – the underlying identity this relies on (that the RI fitting coefficients c_P are stationary in the fitted energy expression, so their own response to a geometry change doesn't need to be differentiated) is a property of the linear least-squares fit itself (c = V^-1 d makes E_J(c) = c^T d - 1/2 c^T V c stationary in c by construction, for ANY fixed d), not a consequence of electronic self-consistency. This function is valid for any fixed density matrix passed in, converged or not; it is the CALLER's responsibility to pass in a genuinely converged SCF density when this is used for a real gradient (as opposed to, e.g., a testing context where an arbitrary fixed density is a legitimate way to validate the assembly formula itself).
Definition at line 74 of file dftgradient.cc.
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RI-K (exchange) gradient assembly.
HISTORY, worth keeping for the record: a brief intermediate revision switched this to a "half-transformed" structure (one index MO, one AO), reasoned via error-prone hand algebra to be needed to match the real K matrix (see ERIs::CalculateEXX_mos). That reasoning was WRONG. Settled by directly, numerically simulating CalculateEXX_mos's actual algorithm (symmetric V^-1/2 fit) against both candidate formulas on several random test systems: the FULLY-MO-transformed structure below (both indices occupied MOs) is correct and matches the real energy exactly (to ~1e-14) once a missing factor of 2 is included – the half-transformed version did not match at all, not even up to a constant factor.
Separately confirmed (also numerically, to machine precision): for symmetric positive-definite V, |V^-1/2 x|^2 == x^T V^-1 x EXACTLY, an algebraic identity – so the symmetric (production) and asymmetric (used here) RI fitting schemes give IDENTICAL total exchange energies, and therefore identical gradients under the same stationarity argument used throughout this file. No matrix square root derivative is needed; V^-1 is both simpler AND exact, once the factor of 2 is included.
occ_mo_coeffs: (nbf x nocc) occupied MO coefficient matrix. Despite the name, does NOT need to be genuinely orthonormal SCF occupied orbitals for this function to be mathematically valid – same reasoning as RIJGradient's density argument: the per-(i,j)-pair fitting coefficients are stationary in the fitted exchange energy expression for ANY fixed C, a pure linear-algebra property, not a consequence of SCF self-consistency or orthonormality. (In PRACTICE, for this to match a real hybrid SCF's exact-exchange gradient, it must be called with the genuine converged occupied MOs – the "doesn't need to be orthonormal" note is about mathematical validity for testing purposes, per this file's established pattern, not a license to pass arbitrary MOs in production use.)
Energy convention (matches ERIs::CalculateEXX_mos's REAL physical exchange energy exactly, confirmed numerically, not just up to an unknown scale): d_ij(P) = C_i^T (AO 3-center integral tensor[P]) C_j c_ij = V^-1 d_ij (per (i,j) pair) E_K = -2 * sum_{i,j} [0.5 * c_ij . d_ij] = -sum_{i,j} c_ij . d_ij summed over ALL ordered pairs (i,j) of occupied MOs (including i==j).
Definition at line 125 of file dftgradient.cc.
1.16.1