Electronic wave function for the

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

I. 33 A. The electronic wave function for the normal benzene molecule can be composed of terms corresponding to the Kekute structures I and... 

Since the >4, representation is present, it follows that these transitions are allowed by the symmetry of the purely electronic wave functions. For the A2 T2 transition we must consider the direct product A2 X T2 x 7, which reduces as follows ... 

The electronic wave function for the n-th state of the complex is written then as the antisymmetrized product of the wave functions of the electron groups introduced above ... 

Both of these expressions are defined in the molecule-fixed (q = 0) coordinate system, and are expectation values over the electronic wave function for the vibronic state r. The contributions of (8.214) are included in the expressions for the first-order energies of the rotational levels given in table 8.6. There are, of course, many non-zero matrix elements in the case (b) basis, all of which are listed by Chiu [40]. 

In the HL treatment of H2 it is assumed that the electronic wave function for the system at large internuclear separations is the product of two unmodified electronic wave functions (Is) of a free H-atom in its ground state, i.e. 

From the determination of the electronic wave function for the time-independent Hamiltonian, Hq + Wqm/cM/ we obtain the reference state of the quantum mechanical subsystem, 0),... 

Fig. 6 Structure of the electronic wave function for the two-site E e system at half-filling. This structure schematically represents the intra-site singlet state . Double-sided arrows indicate the connection of the matrix elements of the wave function via the e coupling...

We call each of these exact one-electron wave functions a molecular orbital (MO), just as we called the exact one-electron wave functions for the hydrogen atom AOs. These exact MOs play a fundamental role in the quantum description of chemical bonding. 

Computer calculations of molecular electronic structure use the orbital approximation in exactly the same way. Approximate MOs are initially generated by starting with trial functions selected by symmetry and chemical intuition. The electronic wave function for the molecule is written in terms of trial functions, and then optimized through self-consistent field (SCF) calculations to produce the best values of the adjustable parameters in the trial functions. With these best values, the trial functions then become the optimized MOs and are ready for use in subsequent applications. Throughout this chapter, we provide glimpses of how the SCF calculations are carried out and how the optimized results are interpreted and applied. 

This comparison of LCAO with VB methods is most easily seen by explicitly writing out the electronic wave functions for the specific case of H2 constructed using both methods. 

Both electrons occupy this bonding orbital, satisfying the condition of indistin-guishability and the Pauli principle. Recall from Section 6.2 that the electronic wave function for the entire molecule in the LCAO approximation is the product of all of the occupied MOs, just as an atomic wave function is the product of all occupied Hartree orbitals of an atom. Thus, we get... 

Now we can compare the LCAO and VB versions of the electronic wave functions for the molecule directly by multiplying out iAmo rearranging terms to obtain... 

It has been known for a long time, especially from the work of Dirac [15] and of Lowdin [78], that the (now idempotent) 1 DM is sufficient to determine the N-electron wave function for the case of a single Slater determinant. It has been equally clear to many workers in the field that such knowledge of the 1DM cannot be adequate to reconstruct the AT-body wave function for the fully interacting electron system, without appeal to the total Hamiltonian. 

The theory of the chemical bond is one of the clearest and most informative examples of an explanatory phenomenon that probably occurs in some form or other in many sciences (psychology comes to mind) the semiautonomous, nonfundamental, fundamentally based, approximate theory (S ANFFBAT for short). Chemical bonding is fundamentally a quantum mechanical phenomenon, yet for all but the simplest chemical systems, a purely quantum mechanical treatment of the molecule is infeasible especially prior to recent computational developments, one could not write down the correct Hamiltonian and solve the Schrodinger equation, even with numerical methods. Immediately after the introduction of the quantum theory, systems of approximation began to appear. The Born Oppenheimer approximation assumed that nuclei are fixed in position the LCAO method assumed that the position wave functions for electrons in molecules are linear combinations of electronic wave functions for the component atoms in isolation. Molecular orbital theory assumed a characteristic set of position wave functions for the several electrons in a molecule, systematically related to corresponding atomic wave functions. 

As noted already, the starting point of the analysis is an approximation to the total electronic wave function for the solid. In particular, we seek a wave function of the form... 

The electronic wave function for the i-th stationary state is only tied to the external sources of the potential aoi, namely, it is independent of the point chosen in the nuclear configuration space. Using the completeness of the solutions to eq.(8) one can write ... 

The many-electron wave function for the groundstate g is parametrized via an exponential excitation operator... 

---