3.19. CASSCF and DFT¶

The Complete Active Space Self-Consistent Field (CASSCF) method is effective at describing static correlation effects, which occur in cases such as open-shell molecules or the dissociation of covalent bonds. However, CASSCF is insufficient for describing dynamic correlation and is often too inaccurate for realistic predictions of energies and molecular properties.

Hybrid approaches that combine CASSCF with Density Functional Theory (DFT) have been developed as alternatives to traditional second-order perturbation theory methods like NEVPT2 and CASPT2.

Since ORCA 6.1, the following hybrid CASSCF-DFT methods are available:

Long-range CASSCF short-range DFT (srDFT)
Multi-Configurational Pair Density Functional Theory (MCPDFT)

3.19.1. General Description¶

3.19.1.1. Multi-Configurational Pair Density Functional Theory¶

MCPDFT evaluates the Kohn-Sham DFT energy

\[\begin{split} E &= E^{1} + E^{\text{H}} + E^{\text{xc}}[\rho,\Pi] \\ &= \sum_{pq} h_{pq} D_{pq} + \frac{1}{2} \sum_{pqrs} D_{pq} D_{rs} g_{pqrs} + E^{\text{xc}}[\rho,\Pi] \end{split}\]

using the one-body density matrix (\(D_{pq}\)) and the two-body density matrix (\(d_{pqrs}\)), both obtained from a preceding CASSCF calculation.

To account for spin polarization in open-shell systems and to improve the accuracy in general, the exchange-correlation (xc) energy employs the on-top pair density \(\Pi(\boldsymbol{r})\)

\[ \Pi = \sum_{pqrs} d_{pqrs} \, \phi_p(\boldsymbol{r}) \, \phi_q(\boldsymbol{r}) \, \phi_r(\boldsymbol{r}) \, \phi_s(\boldsymbol{r}) \]

This enables the computation of alpha and beta electron densities:

(3.132)¶\[ \rho_{\alpha/\beta} = \frac{1}{2} \left( \rho \pm \sqrt{ \rho^2 - 2 \Pi } \right) \]

For spatial points \(\boldsymbol{r}\) where dynamic correlation is suppressed (\(\rho^2 \ge 2\Pi\)), well-established exchange-correlation functionals from single-reference DFT can be applied directly using Eq. (3.132). This technique is known as functional translation.[495]

In the original MCPDFT method,[495] regions where \(\rho^2 < 2\Pi\) were approximated by setting \(\rho_{\alpha/\beta} = \frac{1}{2}\rho\). MCPDFT functionals based on this approach are prefixed with t, e.g., tLDA for translated LDA.

However, in many systems, regions with \(\rho^2 < 2\Pi\) contain important information and should not be neglected or approximated. Since \(\rho_{\alpha/\beta}\) becomes complex in these regions, dedicated exchange-correlation functionals and their functional derivatives are required.[496, 497] Using Eq. (3.132) without approximation is called complex functional translation.[497] These functionals are prefixed with ct, e.g., ctLDA.

In ORCA 6.1 the folloing MCPDFT functionals are available:

Table 3.47 List of available MCPDFT functionals.¶
MCPDFT Functional	Description
tLDA	translated local density approximation (defaults to VWN5)
tPBE	translated Perdew, Becke, Ernzerhofer (PBE)
tPBE0	translated PBE0, 75 % PBE exchange, 25 % exact HF exchange
ctLDA	complex translated local density approximation (defaults to VWN5)
ctPBE	complex translated Perdew, Becke, Ernzerhofer (PBE)
ctPBE0	complex translated PBE0, 75 % PBE exchange, 25 % exact HF exchange

Note

The original MCPDFT is not variational. A variational form that minimizes the MCPDFT energy[498] is not yet available in ORCA.

For excited-state MCPDFT methods, see Sec. Excited States.

3.19.1.2. Long-Range CASSCF Short-Range DFT¶

Dynamic electron correlation is most significant when electrons are close together, making it a short-range phenomenon. The srDFT method[499] addresses this by applying DFT to the short-range part of the interaction and CASSCF to the long-range (static correlation) component.

The srDFT energy expression is:

\[ E = \bra{0} \hat{H}^{\text{lr}} \ket{0} + E^{\text{sr-H}} + E^{\text{sr-xc}} \]

where \(\hat{H}^{\text{lr}}\) is the long-range Hamiltonian evaluated with the CASSCF wavefunction \(\ket{0}\), and \(E^{\text{sr-H}}\) and \(E^{\text{sr-xc}}\) are the short-range Hartree and exchange-correlation energies.

Range separation of explicit two-electron interactions for CASSCF and DFT affects two quantities:

the Coulomb operator that is now split into a lr and sr part

\[\begin{split} \frac{1}{r_{12}} &= \hat{g}^{\text{lr}} + \hat{g}^{\text{sr}} \\ &= \frac{\text{erf}(\mu r_{12})}{r_{12}} + \frac{\text{erfc}(\mu r_{12})}{r_{12}} \end{split}\]

leading to range-separated two-electron integrals

\[ g_{pqrs}^{\text{lr}} = \int\int \phi_p(\boldsymbol{r}_1)\phi_q(\boldsymbol{r}_1) \, \hat{g}^{\text{lr}} \, \phi_r(\boldsymbol{r}_2)\phi_s(\boldsymbol{r}_2) \, d\boldsymbol{r}_1 \, d\boldsymbol{r}_2 \]

and \(g_{pqrs}^{\text{sr}}\).
dedicated short-range xc functionals in \(E^{\text{sr-xc}}\) that are damped for an increasing fraction of CASSCF

An empirical short-range damping parameter \(\mu\) controls the balance between the short-range and long-range mixing. Setting \(\mu\) to \(0\) (zero) restores the single-reference Kohn-Sham DFT energy, as appropriate for a state-specific ground-state calculation. In contrast, increasing \(\mu\) to very large values (\(\mu > 100\)) effectively recovers the CASSCF solution.

Unlike MCPDFT, srDFT is variationally minimized with respect to orbital and CI coefficients. Currently, this is only implemented for exchange-correlation functionals dependent on one-electron densities. For singlets, only total electron density \(\rho\) is used; for other spin multiplicities, both \(\rho\), spin-density \(m\), and their derivatives are considered.[500]

After convergence, the final srDFT energies can be re-evaluated using the on-top pair density as in MCPDFT

The following srDFT functionals are available in ORCA 6.1:

Table 3.48 List of available srDFT functionals.¶
MCPDFT Functional	Description
srLDA	short-range LDA (defaults to VWN5)[239, 501]
srPBE	short-range PBE, also known as Goll, Werner, Stoll (GWS) [502, 503]
srPBE0	short-range PBE0, 75 % srPBE exchange, 25 % exact sr HF exchange
sr-ctLDA	complex translated srLDA
sr-ctPBE	complex translated srPBE
sr-ctPBE0	complex translated srPBE0

3.19.2. Computational Aspects¶

In ORCA 6.1 the CASDFT methods MCPDFT and srDFT are implemented in the TRAH-CASSCF module. They can be invoked by adding the desired functional to the standard CASSCF input.

For example, a state-averaged CASDFT calculation on ethylene with \(\pi\) and \({\pi}^*\) orbitals in the active space:

! TRAH
! ctPBE

%casscf
 nel    2
 norb   2
 mult   1
 nroots 3
end

! TRAH
! sr-ctPBE

%method
 RangeSepMu 0.40 # Default value
end

%casscf
 nel    2
 norb   2
 mult   1
 nroots 3
end

Note that the RIJK approximation is not yet available for CASDFT.

Note that translated sr functionals are also available. In that case, complex translation has to be switched off in the %method block

! TRAH
! sr-ctLDA

%method
 CASDFT
  ComplexTranslation false
 end
end

In contrast to single-reference DFT, the RIJ and RIJCOSX approximations are not activated by default. If desired, they must be explicitly given in the ORCA input file:

! TRAH
! ctPBE0
! RIJCOSX def2-TZVP/C DefGrid3

%casscf
 nel    2
 norb   2
 mult   3
 nroots 1
end

3.19.3. Excited States¶

Excited states can be calculated using state-averaged (SA) methods as in standard CASSCF calculations.

For MCPDFT, energies are evaluated from state-specific one- and two-body densities of a converged SA-CASSCF calculation.[495]

For srDFT, an approximate energy is first minimized using the averaged densities. Final srDFT energies are then evaluated from state-specific (SS) densities.[504]

(a) — Fig. 3.41 Three lowest potential energy curves of ethylene twist calculated with various srLDA methods.¶

(b) — Fig. 3.41 Three lowest potential energy curves of ethylene twist calculated with various srLDA methods.¶

As shown in Fig. 3.41, srDFT with SS densities may produce false minima for twisted ethylene. MCPDFT suffers from similar issues.

To mitigate this, one can diagonalize a linearized CI-DFT Hamiltonian that accounts for deviations from averaged densities:[504, 505]

\[\begin{split} E^{\text{CI-DFT}} &\approx \bra{0} \hat{h} \ket{0} + \delta_{1\, \text{srDFT}} \, \bra{0} \hat{g}^{\text{lr}} \ket{0} \\ &+ E^{\text{(sr)H}}[\rho^{\text{SA}}] + E^{\text{(sr)xc}}[\rho^{\text{SA}},\Pi^{\text{SA}}] \\ &+ \int \left. \left( \frac{ \partial E^{\text{Hxc}}[\rho,\Pi]}{\partial \rho} \right) \right|_* \, \Delta \rho \, d\mathbf{r} + \int \left. \left( \frac{ \partial E^{\text{xc}}[\rho,\Pi]}{\partial \Pi} \right) \right|_* \, \Delta \Pi \, d\mathbf{r} \end{split}\]

As shown in Fig. 3.41, the CI-DFT variant (e.g., sr-ctLDA) resolves the false-minimum problem and yields the best agreement with the highly correlated MRCI+Q method.

The CI-DFT approach can be specified via the simple keyword input

! TRAH
! CI-sr-ctPBE

%casscf
 nel    2
 norb   2
 mult   1
 nroots 3
end

or via the method block

! TRAH
! sr-ctPBE

%method
 CASDFT
  OrthogonalCI true
 end
end

%casscf
 nel    2
 norb   2
 mult   1
 nroots 3
end

In ORCA 6.1 the following CI-DFT keywords are available:

Table 3.49 List of available CI-DFT keywords.¶
CI-DFT Keyword
CI-ctLDA
CI-ctPBE
CI-ctPBE0

CI-srLDA
CI-srPBE
CI-srPBE0

CI-sr-ctLDA
CI-sr-ctPBE
CI-sr-ctPBE0