First order response of the wavefunction

The first order response of the wavefunction is given by the first order response equation (49). At the Hartree-Fock level the response equation can be expressed as 

Whereas 5 1 at the closed-shell Hartree-Fock level of theory is simply the negative of the identity matrix (218), the Hessian is in principle a full matrix with the general structure of (215) and explicit elements given by 

The dimensionality of the Hessian generally does not allow its explicit constmction. The response equation is therefore normally solved in an iterative manner by expanding the solution vector X/i(a)) in a set of trial vectors 

From (218) it is clear that the reduced quantity 5 has a very simple structure = — yBjBy. The reduced Hessian E is constructed in two steps The first and time-consuming step, dominating the calculation of the first order response, is the construction of the 7 vector 

A key to computational efficiency is the realization that the a vector can be formulated as the constmction of modified Fock matrices, as we shall see later 

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For the second derivatives, the first-order response of the wavefunction cannot be eliminated. Further differentiation of Eq. (5) yields... 

We conclude that for a fully variational wavefunction only the first-order response of the wavefunction dX/dx is required to calculate the energy to second order. In particular, the second-order response of the wavefunction d X/dx is not needed for the evaluation of the molecular Hessian. 

These equations are known as the response equations, since they determine the first derivatives (i.e, the first-order responses) of the wavefunction to the perturbation. 

We have established that for a fully variational wavefunction we may calculate the molecular gradient from the zero-order response of the wavefunction (i.e., from the unperturbed wavefunction) and the molecular Hessian from the first-order response of the wavefunction. In general, the 2n + 1 rule is obeyed for fully variational wavefunctions, the derivatives (responses) of the wavefunction to order n determine the derivatives of the energy to order 2n -i- 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wavefunction to first order, but that the calculation of the energy to fourth order requires a knowledge of the wavefunction response to second order. These relationships are illustrated in Table 1. 

The first term is a simple expectation value of the Hamiltonian (differentiated twice with respect to the molecular geometry), whereas the second term contains the first-order response of the wavefunction to the perturbation ... 

Naively, one would expect that second hyperpolarizabilities y are theoretically and experimentally more difficult to obtain than first hyperpolarizabilities (3. From a computational point of view the calculation of fourth-order properties requires, according to the 2n + 1-rule, second-order responses of the wavefunction and thus the solution of considerably more equations than needed for j3 (cf. Section 2.3). However, unlike (3 the second dipole hyperpolarizability y has two isotropic tensor... 

Since second hyperpolarizabilities depend in addition to the first-order also on the second-order response of the wavefunction, the minimal requirements with respect to the choice of basis sets are for y somewhat higher than for the linear polarizabilities a and the first hyperpolarizabilities j8, in particular for atoms and small molecules. For the latter at least doubly-polarized basis sets augmented with a sufficient number of diffuse functions (e.g. d-aug-cc-pVTZ or t-aug-cc-pVTZ) are needed to obtain qualitatively correct results. Highly accurate results at a correlated level will in general only be obtained in quadruple- or better basis sets. 

The first-order correction can be thought of as arising from the response of the wavefunction (as contained in its LCAO-MO and Cl amplitudes and basis functions %v) plus the response of the Hamiltonian to the external field. Because the MCSCF energy functional has been made stationary with respect to variations in the Cj and Cj a amplitudes, the second and third terms above vanish ... 

The first contribution is an expectation value of the second-order perturbation operator with the unperturbed wavefunction of the system. It is negative and is called the diamagnetic contribution The second contribution involves either the first derivative of the perturbed wavefunction, the first-order correction to the wavefunction, a sum over all the other unperturbed states or a linear response... 

So far, the possibility of optimizing the orbitals in the presence of a perturbation (i.e. of making self-consistent property calculations) has been considered only at the Hartree-Fock level. In many cases, however, it is necessary to use a many-determinant wavefunction, either because the IPM ground state is degenerate or because electron-correlation effects are too important to be ignored and it is then desirable to optimize both Cl coefficients and orbitals as in MC SCF theory (Section 8.6). To formulate the perturbation equations, both coefficients and orbitals will be expanded in terms of a perturbation parameter and the orders will be separated the zeroth-order equations will be the MC SCF equations in the absence of the perturbation, while the first-order equations will determine the (optimized) response of the wavefunction, and will thus permit the calculation of second-order properties. Important progress had been made in this area (Jaszunski, 1978 Daborn and Handy, 1983), for particular types of perturbation and Cl function. In fact, however, the equations in their most general form have been known for many years (Moccia, 1974), and are implicit in the stationary-value... 

Because of the separation into a time-independent unperturbed wavefunction and a time-dependent perturbation correction, the time derivative on the right-hand side of the time-dependent Kohn-Sham equation will act only on the response orbitals. From this perturbed wavefunction the first-order response density follows as ... 

If, as is common, the atomic orbital bases used to carry out the MCSCF energy optimization are not explicitly dependent on the external field, the third term also vanishes because (8%v/aA)o = 0. Thus for the MCSCF case, the first-order response is given as the average value of the perturbation over the wavefunction with A=0 ... 

The wavefunction of an IV-electron system is completely characterized by N and the external potential, v(r), because these two quantities fix the Hamiltonian of the system. The electronegativity (x) and the hardness (17) measure the response of the system when N changes at fixed v(r). Within DFT, they are defined as the following first-order [14] and the second-order [15] derivatives,... 

The comparison with Eqs. (3.34) and (3.37) also shows that the linear response function will only give the contributions to the second-order properties that depend on the first-order wavefunction and first-order perturbation Hamiltonian. The contributions that are expectation values of must be obtained by choosing our operator P to be the perturbation-dependent operator P T defined in Eqs. (3.8) to (3.10). The expansion of the expectation value, Eq. (3.109), then becomes... 

Since we can no longer manage without the first-order response, let us consider its evaluation. We have already noted that the variational conditions (equation 16) determine the dependence of the wavefunction on, Differentiating these conditions with respect to x and applying the chain rule, we obtain... 

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],... 

However, because the atomic basis orbitals are attached to the centers, and because these centers are displaced in forming V, it is no longer true that (a%v/aX)o = 0 the variation in the wavefunction caused by movement of the basis functions now contributes to the first-order energy response. As a result, one obtains... 

The basis set representing the first order perturbed orbitals should also be chosen such that it satisfies the imposed finite boundary conditions and can be represented by a form like Equation (36) with the STOs having different sets of linear variation parameters and preassigned exponents. The coefficients of the perturbed functions are determined through the optimization of a standard variational functional with respect to, the total wavefunction . The frequency dependent response properties of the systems are analyzed by considering a time-averaged functional [155]... 

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