Polyelectronic method

Formalism and polyelectronic methods in the framework of the state-specific approach... 

The method of assuring the antisymmetry of a system of electrons, asjfpr example in a polyelectronic atom, is to construct what is often called the Slater determinant.1 If the N elections are numbered 1,2,3,... and each can occupy a state a, b, c,.... the determinant... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

T. Mercouris, Y. Komninos, C.A. Nicolaides, The state-specific expansion approach to the solution of the polyelectronic time-dependent Schrodinger equation for atoms and molecules in unstable states, in C.A. Nicolaides, E. Brandas (Eds.), Unstable States in the Continuous Spectra, Part I Analysis, Concepts, Methods, and Results, Vol. 60 of Advances in Quantum Chemistry, Academic Press, 2010, pp. 333 405. 

Hartree s method was to write a plausible approximate polyelectronic wavefunc-tion (a guess ) for an atom as the product of one-electron wavefunctions ... 

At the simple MO level, the ground state of H2 is described by LMO, in which the bonding a MO is doubly occupied. Expansion (see Chapter 4 for a general method in the polyelectronic case) of this MO determinant into its AO determinant constituents leads to Equation 3.59 (again dropping normalization constants) ... 

A schematic of the SCF method for obtaining the orbitals of a polyelectronic atom. 

The analytical expressions for these m.o.s are complicated. In addition, this method of solving the wave equation is not viable for the more common polyelectronic molecules. That is why another approach, although approximate, is described next. 

At a more theoretical level, nucleophilic reactivity can be treated by the polyelectron perturbation method (25), which in its simplest form gives the perturbation energy in terms of a first-order (Coulombic) term and a second-order term involving orbital energies aand ak (Figure 2). Subsequently, this equation was adapted by Hudson and Filippini (27) and independently by Klopman et al. (28) to the problem of enhanced nucleophilic reactivity. This problem is particularly thorny, and it will now be discussed in some detail. 

Indeed, the techniques of repeated diagonalization of the Hamiltonian on a large basis set and the search for resonance states from the numerical features of the roots upon variation of the function space may lead to reliable conclusions for models or for certain cases of ground shape resonances or of low-lying states of simple systems for which the chosen function space happens to describe the inner part of the resonance accurately. However, it was evident in the late 1960s that such methods had (have) certain limitations when it comes to the MEP in complex, polyelectronic systems. For example, for complex spectra of polyelectronic systems the root-identification criteria may not always lead to physically correct results, even qualitatively. Thus, in spite of careful (and computationally costly) examination, resonance states may be missed or roots may be wrongly attributed to resonances that do not exist. 

Furthermore, and most important, the CCR method is not suitable for the solution of the MEP, just like the direct diagonalization of H(r) on a single set of basis functions is not a practical method for solving the Schrodinger equation for even the ground states of polyelectronic atoms or molecules (More discussion is given in Sections 7 and 8.)... 

The proposal and demonstration in Refs. [11,37] was that, given the correspondence decaying state -o- resonance state, the MEP of the field-free polyelectronic Hamiltonian, H, for the computation of ho/ is best solved by adjusting and adapting formalism and advanced computational methods that were in the process of being developed in the 1960s (and are still used) for the lowest-lying discrete states. 

In conclusion, the discussion, the results, and the accompanying references of fhis chapter have documented, explained, and justified a set of formalisms and methods that can be used for the ab initio computation of fime-independenf and fime-dependent polyelectronic wavefunctions and for fhe quanfifafive undersfanding of basic quantities connected to a variety of field-free and field-induced unstable states (resonances) in the continuous spectra of afoms and molecules. 

Th. Mercouris, Y. Komninos, S. EHonissopoulou, C.A. Nicolaides, Computation of strong-field multiphoton processes in polyelectronic atoms. State-specific method and application to H and LC, Phys. Rev. A 50 (1994) 4109. 

In a series of papers since 1993-1994, we have demonstrated that it is possible to solve quantitatively a variety of TDMEPs in atoms and small molecules, by expanding the nonstationary in terms of the state-specific wavefunc-tions for the discrete and the continuous energy spectrum of the unperturbed system. This SSEA to the solution of the TDSE bypasses the serious, and at present insurmountable, difficulties that the extensively used "grid" methods have, when it comes to solving problems with arbitrary polyelectronic, ground or excited states. Furthermore, it allows, in a transparent and systematic way, the monitoring and control of the dependence of the final resulfs on the type and number of fhe sfafionary states that enter into the expansion that defines fhe wavepackef 4>(f). 

The Schrodinger equation is not exactly soluble for polyelectronic systems due to the electron-electron interaction. In order to solve approximately the Schrodinger equation, different methods are available. The most traditional way is based on the Hartree-Fock (HF) method. An alternative treatment is based on the Density-Functional Theory (DFT). 

When considering the polyelectronic picture, states (or terms) must be used to describe the actual arrangements of the electrons in the valence orbitals. Each term will have a different energy, although it may be composed of a degenerate set of microstates. The terms for the free ion can be determined using methods developed in Chapter 4. 

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