First-order perturbed wavefunction

Previously, Kirkwood(8) had suggested another choice he deduced the first-order perturbed wavefunction from the unperturbed one which was multiplied by a linear combination of the electronic coordinates, i.e. 

The idea to combine a method only polynomial (Eq.6 with 5 0 and c = 0) with the SCF-Cl procedure (Eq.5 with initially developed for the calculation of magnetic observables (9) and later for the electric ones (10). Thus, the first-order perturbed wavefunction is given by ... 

Stevens and Lipscomb s method, for diatomics, avoids the self-consistency error by writing the first-order perturbed wavefunction in terms of the Ji-orbitals, since the ground-state wavefunction involves only [Pg.96]

To do so, we would need a knowledge of the first-order perturbed wavefunction with respect to all atomic displacements. However, there are 3N such perturbed wavefunctions for a molecule with N atoms. Does this mean we have to do 3N calculations in addition to the energy to search a PES Not necessarily. 

The computational cost of the transformation in Eq. (9) is negligible compared to the cost of calculating the generalized polarizabilities, a. Compared to the method used by Geerlings et al. [29], the advantage of our method is that its resolution, i.e., the size of the sine basis, can be defined by the user and is not given by the atomic orbital basis set. The main features of the DRE do not require high resolution. The number of the expansion (sine) basis is proportional to the molecular volume and at a constant resolntion scales linearly with molecular size. This quantity determines the main computational cost, the calculation of the first-order perturbed wavefunctions by the CPHF or CPKS procedure... 

Following Stephens, we can write the vibronic wavefunction as a combination of a pure Born-Oppenheimer zero-order wavefunction, and the first-order perturbed wavefunction due to the nonadiabatic coupling matrix elements as... 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

The first- and second- order RSPT energy and first-order RSPT wavefunction correction expressions form not only a useful computational tool but are also of great use in understanding how strongly a perturbation will affect a particular state of the system. By... 

First of all, consider the parity of the integrands. In the first term onihe right-hand side of Eq. (39) both wavefunctions are either odd or even, thus their product is always even, while x3 is of course odd. The integral between symmetric limits of the resulting odd function of x vanishes and this term mates no contribution to the first-order perturbation. On the other hand the second term is different from zero, as x4 is an even function. 

However, in the first excited state the degree of degeneracy is equal to four. Hence, the first-order perturbation calculation requires the application of Eq. (62). The wavefunctions for the first excited state can be written in the form... 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

Perhaps the simplest and most cost-effective way of treating relativistic contributions in an all-electron framework is the first-order perturbation theory of the one-electron Darwin and mass-velocity operators [46, 47]. For variational wavefunctions, these contributions can be evaluated very efficiently as expectation values of one-electron operators. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux programme87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups an inner core (Is up to 3d), an outer core (4s—4/), and the valence orbitals (5s—6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant ( frozen core ) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately via first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. 

First order terms in Eq. (1) due to vibronic coupling may in general give rise to changes of the electronic wavefunctions. It can be easily seen that eigenfunctions (q) of the zero-order Hamiltonian H(q,0) may be intermixed by first order perturbation yielding... 

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]... 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

Using expansion (16.2) for the wavepacket in terms of the stationary wavefunctions we can derive a set of coupled equations for the expansion coefficients au(t) similar to (2.16). In the limit of first-order perturbation theory [see Equation (2.17)] the time dependence of each coefficient is then given by... 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

If the A wavefunction components of the (unperturbed) excited states Aiu and are Ai(A u))0 and Ai(Ei))0, respectively, the lower lying Ai can be approximated by first order perturbation theory ... 

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