Linear-response wavefunction

The actual form of the Hamiltonian operator hp does not have to be defined at this moment. As in standard perturbation theory, it is assumed that the solution of the electronic structure problem of the combined Hamiltonian HKS +HP can be described as the solution y/(0) of HKS, corrected by a small additional linear-response wavefunction /b//(,). Only these response orbitals will explicitly depend on time - they will follow the oscillations of the external perturbation and adopt its time dependency. Thus, the following Ansatz is made for the solution of the perturbed Hamiltonian HKS +HP ... 

For the non-variational CC wavefunctions, the polarizability is the negative of the linear-response function and may therefore be identified as the second... 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

In the MCSCF linear response theory [48], also called multiconfigurational RPA [54], the reference state is approximated hy a MCSCF wavefunction... 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

The next section (Sect. 2) is devoted to a lengthy discussion of the molecular hypothesis from the point of view of quantum field theory, and this provides the basis for the subsequent discussion of optical activity. Having used linear response theory to establish the equations for optical activity (Sect. 3), we pause to discuss the properties of the wavefunctions of optically active isomers in relation to the space inversion operator (Sect. 4), before indicating how the general optical activity equations can be related to the usual Rosenfeld equation for the optical rotation in a chiral molecule. Finally (Sect. 5), there are critical remarks about what can currently be said in the microscopic quantum-mechanical theory of optical activity based on some approximate models of the field theory. 

An illustrative example of the comparison between the vertical (nonequilibrium) absorption energy obtained with the standard PCM-linear response, its corrected version, and with the wavefunction State-Specific approach based is reported in Table 7-5. 

Both transition energies and oscillator strengths are needed for determination of optically allowed absorption spectra. In the multi-configuration version of the linear response theory (MCLR) one constructs an approximation to the exact linear response function by exposing the optimized (MC) SCF wavefunction 0> to a time-dependent perturbation. In this case the time-dependent wave function assumes the form... 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

The linear response functions X Y a>x described in the previous Chap. 3 provide direct information about the transition properties of the unperturbed molecular solutes. The poles (cok) of X Y a>x correspond to the transition energies from the ground state while the residues determine the associated transition matrix elements. However, at variance with the case of an isolated molecule, the excitation energies from linear response functions of a molecular solute do not correspond to the excitation energies obtained as differences of the energies of the excited states described explicitly by a Cl-like wavefunction expansion [1-3],... 

We note that the atomic axial tensor cannot be expressed directly as an energy derivative since it involves the overlap over two perturbed wavefunctions. However, the perturbed wavefunctions can be obtained as the dot product of the solution vectors N defined by the linear response function in Eq. 2.26 for a magnetic field and nuclear displacement perturbation, respectively. It is important here to note that some care needs to be exercised in the way the orbital rotations, in the language of Section 2.1.1, are defined [253, 254]. 

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