Wavefunction correction

The wavefunctions in Eq. (2.34) are different from the wavefunctions of the free tip and free sample. The effect of the distortion potential (V = Us — Uso and V = Us - Uso), can be evaluated through time-independent perturbation. In the following, we present an approximate method based on the Green s function of the vacuum (see Appendix B). To first order, the distorted wavefunction i)i is related to the undistorted one, i]jo, by 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

In the region z Zn, the integrand vanishes rapidly. The integration in region r a gives 

For z zo, the distorted wavefunction has the same exponential dependence on z as the free wavefunction. Thus, d f dz gains the same factor. The tip wavefunction x gains a similar factor. Therefore, the transmission probability becomes 

We will test this method with the exact solutions of the square-barrier problem in the following subsection. 

---

Ek l and wavefunction corrections /k are expressed in terms of integrals over the... 

Given this particular choice of H, it is possible to apply the general RSPT energy and wavefunction correction formulas developed above to generate explicit results in terms... 

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... 

Combining equations (36) and (38) it can be easily found that the n-th order wavefunction correction is given by [18] ... 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

Using expression (52) into equation (49), after some straightforward manipulation, one can obtain the equivalent rule in order to construct the n-th order wavefunction correction ... 

For example, the first-order wavefunction correction VP1 (i.e., P = + P1 through first order) is given by ... 

The secular determinantal equation is set up in the usual manner, the wavefunctions corrected for the crystal-field interaction are used in the perturbation treatment, energies are generated, and these are used in conjunction with the secular equations to generate new wavefunctions that have now been corrected for spin-orbit coupling. These corrected wavefunctions are used for the calculation of the Zeeman effect. 

Section III then introduces the various approximate energy expressions that are used to determine the wavefunction corrections within each iteration of the MCSCF optimization procedure. Although many of these approximate energy expressions are defined in terms of the same set of intermediate quantities (i.e. the gradient vector and Hessian matrix elements), these expressions have some important formal differences. These formal differences result in MCSCF methods that have qualitatively different convergence characteristics. 

The relation between the wavefunction corrections determined by the WSCI and PSCI methods and the corresponding Newton-Raphson methods may be determined using the matrix partitioning approach. For example, the solution to the PCI matrix eigenvalue equation is equivalent to the solution of the linear equation... 

For relatively small CSF expansion lengths and orbital basis sets, the blocks of the Hessian matrix and gradient vector may be explicitly computed and the appropriate equation may be solved directly to determine the wavefunction corrections for the subsequent MCSCF iteration. For example, the solution of the linear equations required for the WNR and PNR methods may use the stable matrix factorization methods found in the LINPACK library . The routines from this library are often available in efficient machine-dependent assembly code for various computers. Even if this is not the case, the FORTRAN versions of these routines use the BLAS library, resulting in the efficient execution of the primitive vector operations. Similar routines are also available for the direct solution of the eigenvalue equations required for the WSCl and PSCI methods. 

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . 

One promising approach to the problem of effectively reducing the number of direct Cl matrix-vector products is the approximate Hamiltonian operator method of Werner and coworkers - described in Section III. This is an extended micro-iterative method in which the Hamiltonian operator is allowed to be approximated during the solution of the wavefunction correction vector within an MCSCF iteration. 

In relativistic Hartree-Fock calculations a wavefunction correct to 0(c ) yields a total energy correct to 0 c ), but orbital energies (which anyway have no rigorous physical meaning) only correct to 0 c ) [17, 18]. 

For the photodissociation of a triatomic molecule, the asymptotic form of the final state, continuum wavefunction, correctly normalised on the energy scale[42], may be written as[39] ... 

Because of the structure of the matrix element appearing in Eq. (3.48) it is not possible for terms such as k k to contribute directly to the RSPT expressions for Ej even though these factors are certainly contained in the exact wavefunction 7> (they will occur as higher order RSPT wavefunction corrections). That is,... 

---