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# Theoretical Background

We briefly state the basic working equations in the following, as far as required to understand the program output. For a more detailed treatment of the theory see refs. [72,16,71,73,74] and refs. therein.

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

 (7.1)

The (real) expansion coefficients mathend000# and mathend000# are conveniently gathered in a super-vector''

 (7.2)

on mathend000#, the linear space of products of occupied and virtual ground state MOs
mathend000# plus their complex conjugates. mathend000# and mathend000# describe the first-order change of the ground state MOs due to an external perturbation which is represented by mathend000# on mathend000#. For example, if an oscillating electric dipole perturbation along the mathend000# axis is applied, mathend000#, where mathend000# is the electric dipole operator.

Next we define the mathend000# super-matrices''

 (7.3)

where the four-index quantities mathend000# and mathend000# are the so-called orbital rotation Hessians''. Explicit expressions for mathend000# and mathend000# can be found, e.g., in ref. [16]. The vector mathend000# is determined as the solution of the TDHF/TDDFT response problem,

 (7.4)

If mathend000# arises from an electric dipole perturbation mathend000#, the electronic dipole polarizability at frequency mathend000# is

 (7.5)

mathend000#. Similarly, if mathend000# is a component of the magnetic dipole moment operator, the optical rotation is [75]

 (7.6)

where mathend000# is the light velocity.

Excitation energies mathend000# 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),

 (7.7)

The corresponding eigenvectors mathend000# 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 mathend000# 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 mathend000#, which decomposes into a singlet and a triplet part, and of the magnetic orbital rotation Hessian mathend000#. Note that mathend000# is diagonal for non-hybrid DFT, while mathend000# 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 mathend000# denotes the ground state energy, mathend000# and mathend000# are the Fock and overlap matrices, respectively, and indices mathend000# run over all, occupied and virtual MOs.

First, mathend000# is made stationary with respect to all its parameters. The additional Lagrange multipliers mathend000# and mathend000# enforce that the MOs satisfy the ground state HF/KS equations and are orthonormal. mathend000# is the so-called mathend000#-vector, while mathend000# turns out to be the excited state energy-weighted density matrix. Computation of mathend000# and mathend000# 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 mathend000#.

TDHF/TDDFT expressions for components of the frequency-dependent polarizability mathend000# can also be reformulated as variational polarizability Lagrangians [77]

 (7.12)

The stationary point of mathend000# equals to mathend000#. The requirement that mathend000# be stationary with respect to all variational parameters determines the Lagrange multipliers mathend000# and mathend000#. All polarizability components mathend000# 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.

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