Wave function excited states

This is a reliable way to obtain an excited-state wave function even when it is not the lowest-energy wave function of that symmetry. However, it might take a bit of work to construct the input. 

Q-Chem also has a number of methods for electronic excited-state calculations, such as CIS, RPA, XCIS, and CIS(D). It also includes attachment-detachment analysis of excited-state wave functions. The program was robust for both single point and geometry optimized excited-state calculations that we tried. 

With regard to the former, one would like to include as many important configurations as possible. Unfortunately, the definition of an important configuration is often debatable. One popular remedy is the full-valence complete active space SCF (CASSCF) approach in which configurations arising from all excitations from valence-occupied to valence-virtual orbitals are chosen. [29] Since this is equivalent to performing a full Cl within the valence space, the full-valence CASSCF method is limited to small systems. Nevertheless, the CASSCF approach using a well-chosen (often chemically motivated) subspace of the valence orbitals has been shown to yield a much improved depiction of the wave function at all points on a potential surface. Furthermore, the choice of an active space can be adjusted to describe excited state wave functions. 

The lowest n/2 orbitals are doubly occupied and used to build the ground-state determinant wave function, while the rest of the orbitals, the virtual orbital set, will be used later to generate excited-state wave functions.The orbital energies and the SCF-MO s of naphthalene are given as an example and shown in Fig. 2. 

In traditional (non-SF) SR excited states models, the excited state wave-functions are parameterized as follows (see Figure 1) ... 

Table 16.2. The principal terms in the first excited state wave function for Rcc cit the energy minimum. The two sorts of tableaux are given.

Table 16.4. The leading tableaux for the first excited state wave function of C2H4 + CH2 at infinite separation.

The MMCC(2,3), CR-EOMCCSD(T), and other MMCC(mA,mij) methods are obtained by assuming that the Cl expansions of the ground- and excited-state wave functions T ) entering Eq. (50) do not contain higher-than-m -tuply excited components relative to the reference T), where niA < rriB < N. In all MMCC mA,mB) approximations, we calculate the ground- and excited-state energies as follows [47-52,61-63,72] ... 

Similarly, the exact excited state wave function, can be written... 

We have presented a practical Hartree-Fock theory of atomic and molecular electronic structure for individual electronically excited states that does not involve the use of off-diagonal Lagrange multipliers. An easily implemented method for taking the orthogonality constraints into account (tocia) has been used to impose the orthogonality of the Hartree-Fock excited state wave function of interest to states of lower energy. 

In order to obtain transition dipoles we make use of an idea proposed by Casida (54) where the excited state wave function Tj is approximated by... 

In terms of computing adiabatic energy differences, if the Bom-Oppenheimer PES for the excited state can be computed, geometry optimization of that state may be carried out using standard techniques. But, as we have been discussing above, we have not yet devised a scheme for computing the excited-state surface, since ground-state orbitals are not appropriate for minimum-determinantal excited-state wave functions. How then to obtain a better excited-state wave function ... 

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. 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

An electronic transition involves excitation of an electron from the ground state wave function to one of the excited state wave functions. An adiabatic excitation is one that involves adjustment of the nuclear geometry to minimize the energy of the excited molecular system. A vertical excitation is one that occurs so rapidly that the ground state geometry does not have time to change. This latter type of excitation is usually adequate for modeling UV-Vis spectra. 

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