Zeroth-Order Hamiltonian¶

The CASSCF calculation provides a multiconfigurational wavefunction that accurately describes static electron correlation. However, it does not include the dynamic correlation arising from excitations outside the active space.

To account for these missing interactions, XMCQDPT treats the CASSCF wavefunction as the reference state and applies perturbation theory.

The Electronic Hamiltonian¶

The exact electronic Schrödinger equation is

\[ \hat{H}\Psi = E\Psi, \]

where

\(\hat{H}\) is the exact electronic Hamiltonian,
\(\Psi\) is the exact many-electron wavefunction,
\(E\) is the exact electronic energy.

Since solving this equation exactly is impossible for most molecular systems, we approximate the Hamiltonian by separating it into two parts.

Partitioning the Hamiltonian¶

In perturbation theory, the Hamiltonian is written as

\[ \boxed{ \hat{H} = \hat{H}^{(0)} + \lambda \hat{V} } \]

where

\(\hat{H}^{(0)}\) is the zeroth-order Hamiltonian,
\(\hat{V}\) is the perturbation operator,
\(\lambda\) is a bookkeeping parameter that is set equal to one after the derivation.

This decomposition allows the difficult electronic problem to be solved in two stages:

Solve the reference Hamiltonian exactly.
Apply perturbation theory to account for the remaining interactions.

The CASSCF Reference¶

Unlike Hartree–Fock based perturbation methods such as MP2, XMCQDPT does not use a single Slater determinant as its reference.

Instead, the reference wavefunction is the optimized State-Averaged CASSCF wavefunction,

\[ \Psi_0 = \sum_I C_I \Phi_I, \]

where

\(\Phi_I\) are the Configuration State Functions (CSFs),
\(C_I\) are the optimized CI coefficients obtained from the CASSCF calculation.

Since this wavefunction already contains all important electronic configurations within the active space, it provides an excellent starting point for perturbation theory.

What Does the Perturbation Represent?¶

The perturbation operator accounts for electron correlation that was omitted during the CASSCF calculation.

These missing contributions arise from excitations involving orbitals outside the active space.

               External Virtual Orbitals
                     ↑
                     │
Inactive Orbitals → Active Space → Virtual Orbitals

Examples include

excitations from inactive orbitals,
excitations into external virtual orbitals,
simultaneous two-electron excitations,
higher-order interactions neglected by the CASSCF wavefunction.

These configurations are far too numerous to include explicitly in the CI expansion. Instead, their effect is estimated perturbatively.

Why Is the Active Space Not Expanded?¶

One might wonder why these additional configurations are not simply added to the CASSCF calculation.

The reason is computational cost.

Including every external excitation in the CI expansion would increase the number of configurations from a few thousand to millions or even billions.

Instead,

CASSCF treats the most important configurations exactly,
XMCQDPT estimates the remaining contributions using perturbation theory.

This provides a highly accurate description of electron correlation at a much lower computational cost.

The Zeroth-Order Eigenvalue Problem¶

The reference Hamiltonian satisfies

\[ \boxed{ \hat{H}^{(0)} \Psi_i^{(0)} = E_i^{(0)} \Psi_i^{(0)} } \]

where

\(\Psi_i^{(0)}\) are the CASSCF reference states,
\(E_i^{(0)}\) are the corresponding CASSCF energies.

These quantities are obtained directly from the converged CASSCF calculation and serve as the starting point for the XMCQDPT procedure.

Connection with the GAMESS Output¶

The zeroth-order states correspond to the CAS-CI reference states printed at the beginning of the XMCQDPT calculation.

Search for

##########################
###   CAS-CI RESULTS   ###
##########################

For the example calculation, the reference energies are

State	Zeroth-Order Energy (Hartree)
S₀	-2043.651377420
S₁	-2043.481805916
S₂	-2043.462406789

These are the eigenvalues of the zeroth-order Hamiltonian and form the reference upon which the perturbation corrections are built.

Key Points¶

XMCQDPT begins with the converged CASSCF wavefunction.
The Hamiltonian is partitioned into a reference Hamiltonian and a perturbation.
The zeroth-order Hamiltonian is solved exactly using the CASSCF wavefunctions.
The perturbation represents dynamic electron correlation arising from configurations outside the active space.
The reference energies printed under CAS-CI RESULTS become the zeroth-order energies used throughout the XMCQDPT calculation.