State-specific wavefunctions

The first theme is the significance of nonorthonormality (NON) between separately optimized state-specific wavefunctions of the form of Eq. (37). One... 

The ls 2s and the ls 2p P° state-specific wavefunctions were included in the many-electron, many-photon (MEMP) matrix (Eq. (45)), as the two Li thresholds. The coupling of the initial (ground) state, 4 0/ through the dipole field and the corresponding excited configurations were as follows ... 

The SSEA, which is the subject of this review, was introduced in 1993-1994 with the purpose of exploring the potential and the efficiency of using state-specific wavefunctions in solving from first principles TDMEPs related to laser pulse-induced properties and phenomena. In a large number of applications since then, it has been demonstrated that this fundamental, yet conceptually simple, approach to such TDMEPs can indeed be realized computationally. 

As variable parameters for testing convergence are used, the number and the type of the state-specific wavefunctions, the time step At, and the overall range in time for the propagation to provide a normalized solution, x t), equal to 1. The criterion of stable normalization is directly related to the overall adequacy of the N-electron basis set. 

STATE-SPECIFIC WAVEFUNCTIONS FOR LOW- AND HIGH-LYING STATES... 

The general form of state-specific wavefunctions for discrete and for resonance sfafes is Refs. [1,42] and references fherein. 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Some aspects of the bonding in molecules are explained by a model called molecular orbital theory. In an analogous manner to that used for atomic orbitals, the quantum mechanical model applied to molecules allows only certain energy states of an electron to exist. These quantised energy states are described by using specific wavefunctions called molecular orbitals. 

The first part of the review deals with aspects of photodissociation theory and the second, with reactive scattering theory. Three appendix sections are devoted to important technical details of photodissociation theory, namely, the detailed form of the parity-adapted body-fixed scattering wavefunction needed to analyze the asymptotic wavefunction in photodissociation theory, the definition of the initial wavepacket in photodissociation theory and its relationship to the initial bound-state wavepacket, and finally the theory of differential state-specific photo-fragmentation cross sections. Many of the details developed in these appendix sections are also relevant to the theory of reactive scattering. 

We can use the wavefunction of Eq. (2.22) to obtain an approximate normalization for the bound Rydberg states. Specifically we require that... 

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. 

In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

It is worth pointing out that the idea of searching directly for a state-specific solution for the wavefunctions of multiply excited states (MES) implies projections on distinct function spaces with separate optimization of some type, thereby avoiding serious problems having to do with the undue mixing of states and channels of the same symmetry. This idea has since been a central element of our analyses and state-specific computations. In fact, in recent years, such concerns have led to appropriate modifications of conventional methods of quantum chemistry, such as perturbation or coupled-cluster. 

The second notion is concerned with aspects of the issue of the a priori identification of ND and D correlations and the choice of the state-specific set of zero-order orbitals and multiconfigurational wavefunctions in terms of which this identification is assumed and implemented. In this context, 1 use examples from published results and from new computations. 

On the other hand, by emphasizing the state-specific calculation of the wavefunctions of initial and final states and by taking into account orbital NON, it is possible to understand semiquantitatively multiple electron excitations in atoms even at the SCF level. Such one-photon excitations may reach doubly, triply, or even quadruply excited unstable states. Given the existing high-energy photon sources, in atoms these are measurable. Two examples of transitions whose oscillator strengths are finite and reasonable even without the inclusion of electron correlation, are as follows ... 

These calculations and findings were made possible in the framework of the analyses that were published in the 1980s, for example. Refs. [60a, 60b, 61, 62]. Accordingly, the zero-order wavefunction was chosen as the direct, state-specific MCHF solution with only the intrashell configurations for each... 

This became possible not only by the state-specific nature of the computations but also by the realization that the natural orbitals produced from hydrogenic basis sets were the same as the MCHF orbitals that are computable for the intrashell states up to about N = 10 - 12. Therefore, for DES with very high N, instead of obtaining the multiconfigurational zero-order wavefunction from the solution of the SPSA MCHF equations (which are very hard to converge numerically if at all), we replaced the MCHF orbitals by natural orbitals obtained from the diagonalization of the appropriate density matrices with hydrogenic orbitals. 

In symbols, the general compass is the form + 0 " of the wave-function of Eq. (8). In principle, the two parts are represented by different function spaces, whose elements and size depend on the problem. The zero-order wavefunction, is normally obtained self-consistently. Its orbitals belong to the state-specific Fermi-sea, see below. 

The introduction in the early 1970s of the concept and the methodology of the Fermi-sea as the zero-order orbital set for the construction of the state-specific multiconfigurational wavefunction played on the themes... 

The earlier sections contain a number of prototypical ("proof-of-principle"-type) examples where these ideas and methods can be used. I point out that the possibility of constructing state-specific and, hence, compact and directly interpretable wavefunctions has allowed the implementation of practical methodologies for the ab initio nonperturbative solution of the complex-eigenvalue Schrodinger equation for unstable states that are created whenflc external fields are included [10] and of the time-dependent Schrodinger equation for various types of problems [17]. 

Because there is a central theme that pervades the formalisms and applications of this work, its framework has been given the generic name of The state-specific approach (SSA) (Nicolaides, int. J. Quantum Chem. 60, (1996) 119 ibid, 71, (1999) 209). According to the SSA, critical to the development of formalism which is physically helpful as well as computationally practical is, first, the choice of appropriate for each problem forms of the trial wavefunctions and, second, the possibility of employing corresponding function spaces that are as specific and optimal as possible for the state and property of interest. A salient feature of the SSA is that it makes the interplay between electronic structure and dynamics transparent. 

At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is = fl I o+Xas (Eq. (4.1) of text), if necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the asymptotic part, Xas, are functions of either the energy (real or complex) or the time. The many-body square-integrable, %, represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at f = 0. its energy, Eo, is real and embedded inside the continuous spectrum, it is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. 

---