The first-order frequency dependent response of the density matrix can be expanded as
Next we define the ``super-matrices''
If arises from an electric dipole perturbation , the electronic dipole polarizability at frequency is
Excitation energies are the poles of the frequency-dependent density matrix response. They are thus the zeros of the operator on the left-hand side of Eq. (7.4),
The full TDHF/TDDFT formalism is gauge-invariant, i.e., the dipole-length and dipole-velocity gauges lead to the same transition dipole moments in the basis set limit. This can be used as a check for basis set quality in excited state calculations. The TDA can formally be derived as an approximation to full TDHF/TDDFT by constraining the vectors to zero. For TDHF, the TDA is equivalent to configuration interaction including all single excitations from the HF reference (CIS). The TDA is not gauge invariant and does not satisfy the usual sum rules , but it is somewhat less affected by stability problems (see below).
Stability analysis of closed-shell electronic wavefunctions amounts to computing the lowest eigenvalues of the electric orbital rotation Hessian , which decomposes into a singlet and a triplet part, and of the magnetic orbital rotation Hessian . Note that is diagonal for non-hybrid DFT, while generally is not. See refs. [76,15] for further details.
Properties of excited states are defined as derivatives of the excited state energy with respect to an external perturbation. It is advantageous to consider a fully variational Lagrangian of the excited state energy ,
First, is made stationary with respect to all its parameters. The additional Lagrange multipliers and enforce that the MOs satisfy the ground state HF/KS equations and are orthonormal. is the so-called -vector, while turns out to be the excited state energy-weighted density matrix. Computation of and requires the solution of a single static TDHF/TDKS response equation (7.4), also called coupled and perturbed HF/KS equation. Once the relaxed densities have been computed, excited state properties are obtained by simple contraction with derivative integrals in the atomic orbital (AO) basis. Thus, computation of excited state gradients is more expensive than that of ground state gradients only by a constant factor which is usually in the range of .
TDHF/TDDFT expressions for components of the frequency-dependent polarizability can also be reformulated as variational polarizability Lagrangians