Electronic excited states Hamiltonian

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. 

Hamiltonians expressed in matrix forms have been extensively employed in the theory of radiationless transitions of electronically excited states of larger molecules (Bixon and Jortner, 1968 Schlag et al., 1971 Freed, 1972 Nitzan et al., 1972 Avouris et al., 1977 Jortner and Levine, 1981 Felker and Zewail, 1988 Seel and Domcke, 1991). 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

The last fundamental aspect characterizing PCM methods, i.e. their quantum mechanical formulation, is presented by Cammi for molecular systems in their ground electronic states and by Mennucci for electronically excited states. In both contributions, particular attention is devoted to the specific aspect characterizing PCM (and similar) approaches, namely the necessity to introduce an effective nonlinear Hamiltonian which describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. 

The SS approach uses a different effective Hamiltonian 77eff (2.2) for each electronic excited state by calculating Va with the corresponding electron density (i.e., density matrix). The nonlinear character of Va is solved through an iterative procedure [13,14], in which at each iteration, the solvent-induced component of the effective Hamiltonian is computed with the apparent charges determined from the standard ASC-BEM equation by exploiting the first order density matrix of the preceding step. 

The pyramidal acetone-like molecules, such as acetone in an electronic excited state, dimethylamine, yield good examples of applications of the local groups to Czv rotor molecules. As in non-planar pyrocatechin, three different local Hamiltonian operators may be considered. 

Acetone in an electronic excited state, in which the oxygen atom is wagging out-of-plane, furnishes an example of such a case. The corresponding Hamiltonian operator is given by expression (84) in which the threefold periodicity of the rotors are introduced. Similarly, the local rNRG may deduced from the expression (85) ... 

The diagonal elements H,(R) represent the individual Hamiltonian of the /th electronic state. The construction of the Hamiltonian was described in the previous section. The off-diagonal elements V,y (R) describe the diabatic electronic coupling between the 2th and/th electronic states. In cases where the electromagnetic field is explicitly treated couplings due to the field also appear as off-diagonal elements. The subscript X represents the electronic ground state, and subscripts 1,2,..., M) represent the electronic excited states. The number of electronically excited states treated in the description of the photodissociation dynamics depends on the system. 

The concept of Frenkel excitons (molecular excitons) [54] provides a good starting point for the description of the electronically excited states of the LH complexes. The respective model Hamiltonian reads in Heitler-London approximation... 

In Eq. (32) Hf and are the vibrational Hamiltonian, wavefunction and energy of the ground electronic state Hf and Ef are their counterparts for the electronic excited state and is an FC overlap integral. Upon... 

One of the methods most frequently used in calculations of electronic excited states is the configuration interaction technique (CI). When combined with semiempirical Hamiltonians the CI method becomes an attractive method for investigations of electronic structure of large organic systems. Undoubtedly, it is the most popular method for calculations of electronic contributions to NLO properties based on the SOS formalism. The discussion of the CI/SOS techniques is presented in Section 4. 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

An external magnetic field or spin-orbit coupling would add a term zcr, with z a constant, to the excited state Hamiltonian (2.2) [1(c), 22]. The strategy outlined below for measuring the adiabatic electronic sign change will be equally well applicable to the complex Berry phase factors occurring in the presence of a term. 

We shall now apply the results of Section II to a specific model system, commonly used in molecular 4WM. We consider a molecular-level scheme for the absorber, consisting of a manifold of vibronic levels belonging to the ground electronic state, denoted a>, c>,..., and a manifold of vibronic levels belonging to an excited electronic state, denoted by, d>,. .., (Fig. 4). The ground and the electronically excited states will be denoted g and e>, respectively. The absorber is further coupled to a thermal bath, and the combined Hamiltonian for the molecule and the bath js -44- 57.58,64... 

We adopt here the common spectroscopic notation and use a superscript double prime and prime to denote quantities belonging to the ground and to the electronically excited states, respectively. Fs and Qj are the conjugate momentum and the normal coordinate of the jth mode of the excited state. The transformation [Eqs. (86) and (87)] defines pj and q, which are the dimensionless momentum and normal coordinate of the jth mode. A similar transformation between pj, qj, and Pj, Q] is defined by changing all prime indexes in Eqs. (86) and (87) to double primes. j) and m are the frequency and the mass of the jth mode. The present Hamiltonian [Eq. (85)] is a special case of the general two-manifold Hamiltonian [Eq. (41)]. We shall now introduce a vector notation and define the N component vectors q and q", whose components are q - and q j, j = 1 respectively. The normal modes... 

We consider a one-photon excitation from the ground electronic state g) to an electronically excited state e). In the absence of the light the Hamiltonian H written within the Born-Oppenheimer approximation is diagonal in the electronic quantum numbers and can be symbolically written as... 

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