Analytic gradients for compressed multistate pair-density functional theory

Dipole Moments and Transition Dipole Moments Calculated by Pair-Density Functional Theory with State Interaction

“…We introduce a Lagrangian that is stationary in wave function variables corresponding to nonredundant orbital rotations X pq and unitary transformations of the CI vectors, such as rotations Y RA between a state within and a state outside the model space and rotations Z RS between states within the model space: L P Q CMS = H P Q CMS + prefix∑ p > q ∂ E SA‐CAS ∂ X p q x p q + prefix∑ R , A ∂ E SA‐CAS ∂ Y R A y R A + prefix∑ R > S ∂ Q normala−a ∂ Z R S z R S where x pq , y RA , and z RS are Lagrange multipliers associated with the derivatives of state-averaged CASSCF energy E SA‑CAS and classical Coulomb energy Q a‑a , and the summations run respectively over orbital indices p and q , over intermediate state indices R and S , and over configuration state functions defined by A . The three sets of Lagrange multipliers are obtained by solving the system of coupled linear equations generated by ∂ L P Q CMS ∂ X p q = ∂ L P Q …”

“…The second and third terms in eq 11 are reminiscent of those in SA-PDFT and CMS-PDFT analytical gradients, 25,26 but involve electric dipole integrals instead of nuclear derivatives of one-electron integrals, in particular:…”

“…We introduce a Lagrangian that is stationary in wave function variables corresponding to nonredundant orbital rotations X pq and unitary transformations of the CI vectors, such as rotations Y RA between a state within and a state outside the model space and rotations Z RS between states within the model space: where x pq , y RA , and z RS are Lagrange multipliers associated with the derivatives of state-averaged CASSCF energy E SA‑CAS and classical Coulomb energy Q a‑a , and the summations run respectively over orbital indices p and q , over intermediate state indices R and S , and over configuration state functions defined by A . The three sets of Lagrange multipliers are obtained by solving the system of coupled linear equations generated by The derivatives in eq take the simple form Using the chain rule, we have where the first term is the Hellmann–Feynman contribution, and the remaining terms are the non-Hellmann–Feynman contributions originating from the nonvariational character of the CMS-PDFT energy.…”

“…When P = Q , the first term is a derivative of a diagonal element given by eq : When P ≠ Q , this derivative takes a different form that follows from eqs and : By nesting outcomes of eqs and into eq , we obtain the nuclear part of the dipole moment and the electronic Hellmann–Feynman contribution The second and third terms in eq are reminiscent of those in SA-PDFT and CMS-PDFT analytical gradients, , but involve electric dipole integrals instead of nuclear derivatives of one-electron integrals, in particular: Here, M is the index of the adiabatic CMS-PDFT state for which the dipole moment is computed, E pq is the one-electron excitation operator, J runs over reference SA-CASSCF states with weights ω, and Λ runs over configuration state functions. Finally, the last term in eq is zero by construction because Q a–a does not depend on the electric field.…”

Linearized Pair-Density Functional Theory

“…Of the various MS-PDFT methods, compressed multi-state PDFT (CMS-PDFT) has been shown to have the best balance of computational efficiency and accuracy for the widest variety of systems, and it computes PESs similar to the more expensive extended multistate complete-active-space 2nd-order perturbation theory (XMS-CASPT2) for a variety of test cases. 28 However, during the development of CMS-PDFT analytical gradients, 29 it was discovered that CMS-PDFT struggles with linear molecules with degenerate 1 ∆ u states.…”