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SC-NEVPT2 Nuclear Gradient

OpenQP computes the nuclear gradient of the single-state, strongly contracted NEVPT2 (SC-NEVPT2) total energy analytically. The implemented energy is the Dyall-Hamiltonian, strongly contracted correction to a state-specific CASSCF reference. Set [input] method=caspt2, [pt2] h0=dyall, contraction=strong, and use a gradient-driven run type.

Unlike CASSCF, SC-NEVPT2 is not variational with respect to the reference orbitals or CI coefficients. Its derivative therefore includes coupled orbital and CI response, the response of the semicanonical inactive and virtual orbitals, and the derivative-integral contraction of the resulting relaxed one- and two-particle densities. The public energy and gradient calls share one CASSCF reference, one second-order evaluation, and one coupled response solve at each geometry.

Selecting the route

[pt2] gradient controls the nuclear-gradient route:

Value Behavior
auto Use the analytic derivative when the input and current reference satisfy its scope; otherwise report the reason and use central differences. This is the default.
analytic Require the analytic derivative. An unsupported input, non-stationary reference, ill-conditioned semicanonical response, or unsolved coupled response is an error.
numerical Force Cartesian central differences of the converged SC-NEVPT2 energy.

The analytic route applies only when all of the following are true:

  • method=caspt2, variant=caspt2, h0=dyall, and contraction=strong select single-state SC-NEVPT2;
  • reference=casscf selects a state-specific, stationary CASSCF reference;
  • the reference is a closed-shell singlet from RHF orbitals, with no density functional and a contiguous inactive/active/virtual partition;
  • the native determinant-space integral backend and compiled nevpt2_gradient entry point are available;
  • no IPEA, real, imaginary, or QDPT energy-denominator shift is requested; and
  • the configured [cas] or [pt2] max_memory can hold the active-space, orbital-space, and native Cartesian density intermediates.

auto does not hide an implementation failure. It falls back only for a declared scope or reference-applicability condition; an unexpected native or Python failure remains an error.

Input example

The concise .oqp request — the SC-NEVPT2 route is the caspt2 model with the Dyall zeroth-order Hamiltonian and strong contraction selected in pt2(...):

caspt2/sto-3g opt(S0) cas(active_electrons=2,active_orbitals=2,frozen_core=1)
pt2(h0=dyall,contraction=strong,gradient=analytic)
geom="""
Li  0.0  0.0  0.0
H   0.0  0.0  1.6
"""

The equivalent legacy .inp spelling is:

[input]
system=
  Li  0.0  0.0  0.0
  H   0.0  0.0  1.6
basis=sto-3g
method=caspt2
runtype=optimize

[cas]
active_electrons=2
active_orbitals=2
frozen_core=1

[ci]
target_spin=singlet

[pt2]
reference=casscf
h0=dyall
contraction=strong
gradient=analytic

[optimize]
istate=0

The runnable input is examples/WF_methods/LiH_SC-NEVPT2_optimize.inp. Single-state SC-NEVPT2 publishes one energy and one gradient row, so [optimize] istate=0 is the public selector. Use [pt2] target_roots to select the physical CASSCF root corrected by the single-state calculation.

The equivalent Python helper exposes the same route:

from oqp.openqp import OpenQP

job = OpenQP("lih_sc_nevpt2_opt", silent=1)
job.molecule("Li 0 0 0; H 0 0 1.6", charge=0, multiplicity=1)
job.nevpt2(active_electrons=2, active_orbitals=2, frozen_core=1,
           contraction="strong", gradient="analytic",
           basis="sto-3g", runtype="optimize",
           casscf={"gradient_norm_tol": "1.0e-10"})
mol = job.run()

Stationarity and response diagnostics

The SC-NEVPT2 Lagrangian assumes a stationary CASSCF reference. Its gradient error is first order in the residual CASSCF orbital-rotation gradient, so [casscf] gradient_norm_tol controls the useful precision of the nuclear gradient. Each analytic evaluation reports:

  • the CASSCF orbital-gradient norm;
  • the smallest semicanonical inactive or virtual orbital-energy gap;
  • the number of exactly degenerate pairs treated as free orbital gauges;
  • the smallest Dyall denominator;
  • the relative residual of the coupled orbital/CI response solve; and
  • the maximum asymmetry of the total orbital Lagrangian.

An exactly degenerate inactive or virtual pair is accepted only when the corresponding Lagrangian numerator also vanishes at working precision. The rotation then is a free gauge and its multiplier is zero. A merely near- degenerate pair remains ill-conditioned and is refused by the analytic route. This distinction permits symmetry-required degeneracies, such as the two virtual pi orbitals of linear LiH, without assigning an unstable finite multiplier.

The native derivative-integral contraction returns the complete relaxed- density gradient. It is not projected through the integral-symmetry petite list, because a state-specific or excited CASSCF root need not furnish a totally symmetric contracted density.

Verification and optimization reference

The analytic derivative is tested against five-point central differences of the SC-NEVPT2 total energy. For LiH/STO-3G with CAS(2,2), a frozen Li 1s core, and exactly degenerate virtual pi orbitals, the largest analytic-minus-finite- difference component is 3.4e-11 Hartree/Bohr.

The LiH optimization converges to E(SC-NEVPT2) = -7.8819849582 Hartree and r(Li-H) = 2.9270761417 Bohr. Repeating the calculation reproduces the bond length to 3e-13 Bohr; starting from 1.75 Angstrom reaches the same minimum to 2.5e-7 Bohr.

The former linear H4/STO-3G CAS(2,2) optimization is not a suitable coordinate reference. On that surface H4 dissociates into two H2 fragments separated by about 7.4 Bohr, and their relative translation is nearly free. Independent runs can have the same energy to 12 decimal places while their Cartesian coordinates differ by about 0.067 Bohr. That behavior is a property of the nearly flat dissociation surface, not evidence of a nuclear-gradient defect.

Cost and memory

One analytic evaluation replaces the 6 * natom displaced SC-NEVPT2 energies needed by a two-point Cartesian central difference. Its dense memory estimate includes the orbital-space fourth-rank tensors, active-space RDM and adjoint tensors, the native nbf^4 density copy, and, for a harmonic basis, the simultaneous nbf_cart^2 * nbf^2 and nbf_cart^4 Cartesian expansions. If this estimate exceeds the configured memory ceiling, gradient=auto selects the numerical route and gradient=analytic reports the required memory.