Wave function state and

Within the Bom-Oppenheimer approximation, the last term is a constant. It is seen that the Hamilton operator is uniquely determined by the number of electrons and the potential created by the nuclei, V e, i.e. the nuclear charges and positions. This means that the ground-state wave function (and thereby the electron density) and ground state energy are also given uniquely by these quantities. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

Chapter 14 deals with orbital correlation diagrams following Woodward and Hoffmann [3]. State wave functions and properties of electronic states are deduced from the orbital picture, and rules for state correlation diagrams are reviewed, as a prelude to an introduction to the field of organic photochemistry in Chapter 15. 

The simplest approach, of course, is to maintain the minimum-determinantal description and reoptimize all of the orbitals. In practice, however, such an approach is practical only in instances where die ground-state and the excited-state wave functions belong to different incduciblc representations of die molecular point group (cf. Section 6.3.3). Otherwise, the variational soludon for die excited-state wave function is simply to collapse back to the ground-state wave function And, even if the two states do differ in symmetry, the desired excited state may not be the lowest energy such state widiiii its irrep, to which variational optimization will nearly always lead. 

Methods for generating excited-state wave functions and/or energies may be conveniently divided into methods typically limited to excited states that are well described as involving a single excitation, and other more general approaches, some of which carry a dose of empiricism. The next three sections examine these various methods separately. Subsequendy, the remainder of the chapter focuses on additional spectroscopic aspects of excited-state calculations in both the gas and condensed phases. 

C. A. Coulson The intensity of a vibrational transition depends on the matrix component between initial and final state wave functions, and also on the energy of the transition. The second of these factors leads to a lower intensity for the higher harmonics. The first one is a complicated situation, in which the increased weight of the ionic structures does certainly give an increased /-value but other factors, such as the overlap of the wave functions will lead here (as in usual molecules) to a distinct decrease in intensity. 

In addition to bound-state wave functions and energies, there are continuum wave functions and energies all positive energies are allowed. 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

To understand the development or the absence of reflection structures one must imagine — in two dimensions — how the continuum wavefunction for a particular energy E overlaps the various ground-state wave-functions and how the overlap changes with E. This is not an easy task Figure 9.9 shows two examples of continuum wavefunctions for H2O. Alternatively, one must imagine how the time-dependent wavepacket, starting from an excited vibrational state, evolves on the upper-state PES and what kind of structures the autocorrelation function develops as the wavepacket slides down the potential slope. 

FIGURE 7.Ans.l (a) Resonance structures of pentadienyl radical and their symmetry properties, (b) The VB mixing of the resonance structures, (c) The quasiclassical (spin alternant) determinant that dominates the ground-state wave function, and the corresponding secondary determinants, and the resulting spin density distribution (p) in the ground state, (d) The spin distribution in the covalent excited state. 

This output displays part of the information that is given by the XMVB program at the end of an L-VBCISD calculation on F2. Each fundamental structure is a linear combination of VB functions that possess the same nature in terms of spin-pairing and charge distributions. The coefficients of these VB functions are extracted from the multistructure ground-state wave function, and are renormalized (see Eqs. 9.13-9.14). 

In general however, for any component, rm, of the ground state, and for any Jahn-Teller operator, Qy, we have for the ground state wave function and for the ensuing Jahn-Teller matrix element, a and / being the mixing coefficients,... 

At this point, it is of interest to discuss the relationship between MO theory and the intensity of electronic transitions. The oscillator strength of an electronic absorption band is proportional to the square of the transition dipole moment integral, ( /gM I/e) where /G and /E are the ground- and excited-state wave functions, and r is the dipole moment operator. In a one-electron approximation, (v(/G r v(/E) 2= K Mrlvl/fe) 2> where v /H and /fe are the two MOs involved in the one-electron promotion v /H > v / ,. Metal-ligand covalency results in MO wave... 

In the EOM-CC and LR-CC approaches, excited state wave functions and energies are built on top of a single-determinant CC description of the ground state (or other convenient reference state). Therefore, we begin with an overview of the ground state CC method. 

Equation 6.33 is completely general. For the two-spin system, it results in the transitions we identified in Fig. 6.2, while the double quantum transition between and 4, and the zero quantum transition between 02 and 03 are forbidden. Note that this statement is true for this treatment, which employs stationary state wave functions and time-dependent perturbations, but as we shall see in Chapter 11, it is easy with suitable pulse sequences to elicit information on zero quantum and quantum double processes. For our present purposes in the remainder of this chapter we accept the validity of Eq. 6.33. 

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