Polyelectronic atoms and molecules

A second complication with multi-electron species is that we must accoimt for electron spin. Spin is characterised by the quantum number s, which for an electron can only take the 

It was stated above that the Schrodinger equation carmot be solved exactly for any molecular systems However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as HD+), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefimction for the molecule can be written in the following form  

The total energy equals the sum of the nuclear energy (the electrostatic repulsion between the positively charged nuclei) and the electronic energy. The electronic energy comprises 

When the Born-Oppenheimer approximation is used we concentrate on the electronic motions the nuclei are considered to be fixed. For each arrangement of the nuclei the Schrodinger equation is solved for the electrons alone in the field of the nuclei. If it is desired to change the nuclear positions then it is necessary to add the nuclear repulsion to the electronic energy in order to calculate the total energy of the configuration. 

If we assume that the wavefunctions are normalised then it can easily be seen that the total energy E is the sum of the individual orbital energies E and E2 (E = J dr,0i(ri) i0i(rx) and 2 = 1 dT2(t 2i 2) 2(t 2 2))- When the separation of variables method is used the solutions for each electron are just those of the hydrogen atom (Is, 2s, etc.) in Equation (2.19) with Z = 2. 

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The division into Feshbach and shape resonances in polyelectronic atoms and molecules... 

Given the normal formulation of QM in ferms of sfationary states, including the case of resonances in fhe confinuous specfrum (see the superposition wavefunction of Eq. (2)), and fhe corresponding expressions which are measured on the energy axis, the question arises as to how easy it is to prepare unstable states in polyelectronic atoms and molecules and to observe their time-dependence. The answer involves more than one components. The most frequently mentioned is to consider the magnitude of fhe lifefime (provided, of course, fhaf if is known) relative to the duration of fhe excitation process. Obviously, if fhe former is much longer than the latter, the concept of a decaying sfafe is valid. 

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. 

The problem of the structure of the hydrogen atom is the most important problem in the field of atomic and molecular structure, not only because the theoretical treatment of this atom is simpler than that of other atoms and of molecules, but also because it forms the basis for the discussion of more complex atomic systems. The wave-mechanical treatment of polyelectronic atoms and of molecules is usually closely related in procedure to that of the hydrogen atom, often being based on the use of hydrogenlike or closely related wave functions. Moreover, almost without exception the applications of qualitative and semiquantitative wave-mechanical arguments to chemistry involve the functions which occur in the treatment of the hydrogen atom. 

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 State-Specific Expansion Approach to the Solution of the Polyelectronic Time-Dependent Schrodinger Equation for Atoms and Molecules in Unstable States... 

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. 

When extending the molecular orbital concept developed for the monoelec-tronic species H2 to polyelectronic diatomic molecules, we start by acknowledging the role of two fundamental approximations (a) one associated with the existence of two nuclei as attractive centres, namely the Born-Oppenheimer approximation, as already encountered in H2" and (b) the other related to the concept of the orbital when two or more electrons are present, that is the neglect of the electron coulomb correlation, as already discussed on going from mono- to polyelectronic atoms. Within the orbital approach, an additional feature when comparing to H2" is the exchange energy directly associated with the Pauli principle. 

The Hamiltonian, H, for polyelectronic systems (atom or molecule) is given by Eq. (1.110). It can be divided into three terms HR, a term intrinsic to the radiation field, HM, a term intrinsic to the molecular field, and Hh a term for the interaction between the radiation and the molecular fields. The term //, is further divided into first and second order terms of the vector potential, A, for the quantified operators, p-A = A p, using Vvl = 0. 

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 above rule can readily be extended to other polyelectronic systems, like the tt system of benzene (6), or to molecules bearing lone pairs as in formamide (7). In this latter case, calling n, c, and o, respectively, the tt atomic orbitals of nitrogen, carbon, and oxygen, the VB wave function describing the neutral covalent structure is given by Equation 3.10 ... 

In the above treatment of the hydrogen molecule ion and the hydrogen molecule, the effect of electronic spin has been excluded, but the complete wave function of an electron must include not only the orbital motion, with which we have been concerned so far, but also a contribution for the spin. With single atoms it was possible to introduce a fourth quantum number s in addition to the three quantum numbers , I and m in order to account for the spin of the electron, and for polyelectronic molecules it is possible to proceed in an analogous manner. The complete wave function of an electron is considered to be the product of the orbital wave function, i,e. the wave function that we have been considering so far, and a wave function representing the orientation of the spin axis of the electron. 

As we have seen previously (Chapter 5), the eigenfunction for a polyelec-tronic atom is antisymmetric with respect to the exchange of the coordinates of any two electrons, and can be expressed as a Slater determinant whose elements are the various occupied spin-orbitals (or a linear combination of Slater determinants, in the case of open-shell atoms). The same appfies to polyelectronic molecules, the atomic orbitals being replaced by the various occupied molecular orbitals associated with the a and /3 spin-functions spin molecular orbitals. Thus, for the molecules H2O, NH3 or CH4 having five doubly occupied m.o.s (one core s orbital and four valence m.o.s), we have... 

The importance of the hydrogen molecule ion for the theory of diatomic molecules is similar to the importance of the hydrogen atom for our understanding of atoms both H and H2 are one-electron systems for which the Schrbdinger equation can be solved exactly. The exact solution of the one-electron species is then used as a starting point for the discussion of polyelectron species for which exact solutions of the Schrodinger are unavailable. 

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