Perturbation Expansion¶

After defining the zeroth-order Hamiltonian,

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

the next objective is to determine how the perturbation operator modifies the energies and wavefunctions of the CASSCF reference states.

Since the perturbation is assumed to be much smaller than the reference Hamiltonian,

\[ \hat{V}\ll \hat{H}^{(0)}, \]

the exact solution can be written as a series expansion.

Expansion of the Wavefunction¶

The exact electronic wavefunction is expressed as

\[ \boxed{ \Psi_i = \Psi_i^{(0)} + \lambda\Psi_i^{(1)} + \lambda^2\Psi_i^{(2)} +\cdots } \]

where

\(\Psi_i^{(0)}\) is the CASSCF reference wavefunction,
\(\Psi_i^{(1)}\) is the first-order correction,
\(\Psi_i^{(2)}\) is the second-order correction.

Each successive term improves the description of the electronic state by incorporating interactions with external configurations.

Expansion of the Energy¶

Similarly, the electronic energy is expanded as

\[ \boxed{ E_i = E_i^{(0)} + \lambda E_i^{(1)} + \lambda^2E_i^{(2)} +\cdots } \]

where

\(E_i^{(0)}\) is the CASSCF energy,
\(E_i^{(1)}\) is the first-order correction,
\(E_i^{(2)}\) is the second-order correction.

The total energy is therefore built progressively by adding higher-order corrections to the reference energy.

Substituting into the Schrödinger Equation¶

Starting from

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

and substituting the perturbation expansions gives

\[ \left( \hat{H}^{(0)} +\lambda\hat{V} \right) \left( \Psi^{(0)} +\lambda\Psi^{(1)} +\lambda^2\Psi^{(2)} +\cdots \right) = \left( E^{(0)} +\lambda E^{(1)} +\lambda^2E^{(2)} +\cdots \right) \left( \Psi^{(0)} +\lambda\Psi^{(1)} +\lambda^2\Psi^{(2)} +\cdots \right). \]

Collecting terms of equal powers of \(\lambda\) generates a hierarchy of equations that can be solved sequentially.

Zeroth-Order Equation¶

The zeroth-order terms reproduce the CASSCF eigenvalue problem,

\[ \boxed{ \hat{H}^{(0)} \Psi^{(0)} = E^{(0)} \Psi^{(0)} } \]

This equation has already been solved during the CASSCF calculation.

Its solutions become the reference states for the perturbation treatment.

First-Order Equation¶

The first-order equation is

\[ \boxed{ \left( \hat{H}^{(0)} - E^{(0)} \right) \Psi^{(1)} = - \left( \hat{V} - E^{(1)} \right) \Psi^{(0)} } \]

This equation determines how the perturbation mixes the CASSCF reference state with configurations outside the active space.

The first-order wavefunction therefore contains information about electronic excitations that were not explicitly included in the CASSCF calculation.

Second-Order Energy¶

The quantity of greatest interest in XMCQDPT is the second-order energy correction,

\[ \boxed{ E^{(2)} = \sum_r \frac { \left| \left< r \left| \hat V \right| 0 \right> \right|^2 } { E_0^{(0)} - E_r^{(0)} } } \]

where

\(r\) denotes external electronic configurations,
\(\hat V\) is the perturbation operator,
\(E_r^{(0)}\) are the energies of the external configurations.

This expression estimates the energy lowering produced by electron correlation outside the active space.

Unlike CASSCF, these external configurations are not included explicitly in the wavefunction. Their contributions are evaluated mathematically through perturbation theory.

Why Only Second Order?¶

In principle, perturbation theory can be continued indefinitely,

\[ E = E^{(0)} + E^{(1)} + E^{(2)} + E^{(3)} +\cdots \]

However,

third-order and higher corrections are considerably more expensive,
the second-order correction already recovers a large fraction of the missing dynamic correlation,
second-order perturbation provides an excellent balance between computational cost and accuracy.

For this reason, GAMESS implements the MCQDPT2 method, where only the second-order correction is evaluated.

Connection with XMCQDPT¶

For a single electronic state, the perturbation expansion described above would be sufficient.

In excited-state calculations, however, several electronic states often lie very close in energy and interact strongly.

Treating each state independently may therefore lead to numerical instabilities and inaccurate energies.

XMCQDPT overcomes this limitation by constructing an effective Hamiltonian that treats all reference states simultaneously before applying the perturbation correction.

This multistate treatment is the defining feature of the XMCQDPT method and is discussed in the next section.

Key Points¶

Perturbation theory assumes that the CASSCF wavefunction is a good approximation to the exact solution.
Both the wavefunction and energy are expanded as power series.
The zeroth-order solution corresponds to the converged CASSCF calculation.
The second-order energy correction accounts for dynamic electron correlation outside the active space.
XMCQDPT evaluates only the second-order correction, providing a highly accurate and computationally efficient treatment of electron correlation.