The first-order frequency dependent response of the density matrix can be expanded as

(7.1) |

The (real) expansion coefficients and are conveniently gathered in a ``super-vector''

(7.2) |

on , the linear space of products of occupied and virtual ground state MOs

plus their complex conjugates. and describe the first-order change of the ground state MOs due to an external perturbation which is represented by on . For example, if an oscillating electric dipole perturbation along the axis is applied, , where is the electric dipole operator.

Next we define the ``super-matrices''

(7.3) |

where the four-index quantities and are the so-called ``orbital rotation Hessians''. Explicit expressions for and can be found, e.g., in ref. [16]. The vector is determined as the solution of the TDHF/TDDFT response problem,

If arises from an electric dipole perturbation , the electronic dipole polarizability at frequency is

(7.5) |

. Similarly, if is a component of the magnetic dipole moment operator, the optical rotation is [75]

(7.6) |

where is the light velocity.

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 corresponding eigenvectors are the transition density matrices for a given excitation (also called ``excitation vectors'' in the following). They are required to be normalized according to

(7.8) |

Transition moments are evaluated by taking the trace with one-particle operators, e.g.,

(7.9) |

for the electric and

(7.10) |

for the magnetic transition dipole moments.

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 [72], 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 [16],

(7.11) |

Here denotes the ground state energy, and are the Fock and overlap matrices, respectively, and indices run over all, occupied and virtual MOs.

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 [77]

(7.12) |

The stationary point of equals to . The requirement that be stationary with respect to all variational parameters determines the Lagrange multipliers and . All polarizability components are processed simultaneously which allows for computation of polarizability derivatives at the computational cost which is only 2-3 higher than for the electronic polarizability itself.