Sum-over-States Treatment

In the first approach, the so-called sum-over-states treatment of the corrections due to molecular vibration, the effects of the perturbation on the electronic and vibrational part of the wavefunction are treated simultaneously. This means that perturbation theory as developed in Section 3.2 is applied to the vibronic wavefunctions. In the presence of an external electric field with component p the perturbed vibronic wavefunction for, e.g., the electronic ground state k = 0 and an arbitrary vibrational state V, d ot,(f), is thus obtained through first order, Eq. (3.27), as  

The summation is typically split into two parts, defining the components of the vibrational polarizability, unfortunately sometimes also called atomic polarizability, where the summation goes over all vibrational states v v oi the same electronic state n = 0 

one can make use of the the Born-Oppenheimer approximation, as discussed in Section 2.2, and approximate the unperturbed vibronic wavefunctions as a simple product of vibrational and electronic wave function Rk ), i-e. 

Using this ansatz in the expressions for the polarizabilities one obtains 

The electronic ground-state expectation values in the numerator of the vibrational polarizability are components of the permanent electric dipole moment, Elq. (4.40), and we can therefore write the vibrational polarizability more compactly as 

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A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

The advantage of the sum-over-states treatment is the pictorial nature of the decomposition of the observed B terms into contributions from the pairwise mixing of observable zero-order states. Comparison of a series of molecules can then take advantage of known differences in their relative transition energies, intensities, and polarization directions. 

The high value of the electron density at the nucleus leads to the enhancement of the electron EDM in heavy atoms. The other possible source of the enhancement is the presence of small energy denominators in the sum over states in the first term of Eq.(29). In particular, this takes place when (Eo — En) is of the order of the molecular rotational constant. (It is imperative that a nonperturbative treatment be invoked when the Stark matrix element e z(v /0 z v / ) is comparable to the energy denominator (Eq En) [33].) Neglecting the second term of the right-hand side of Eq.(29), which does not contain this enhancement factor [8, 27], we get... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

As pointed out above, in the time-dependent case the correlation treatment cannot be based on time-dependent Hartree-Fock orbitals - at least not on the real frequency axis in the vicinity of poles of the response functions. Thus, the polarization of the wavefunction must be described through the variables of the correlation method, i.e. for the CC approach by means of the cluster amplitudes. This has important implications on the choice or suitability of correlation methods. As it is apparent from the sum-over-states expression for the -th response function [96]... 

In this chapter we review some of the developments that have been made over the past fifteen years with regard to the calculation of vibrational contributions to linear and nonlinear (NLO) optical properties. Despite a number of advances it is important to recognize that more are needed since there is still no fully satisfactory general treatment for either resonant or non-resonant NLO processes in polyatomic molecules. Two major intertwining approaches to practical computations that include electrical and mechanical anharmonicity have emerged. The older approach is from the viewpoint of ordinary sum-over-states perturbation theory and it is presented in Section 1. The other approach, discussed in Section 2, is from what may be called the nuclear relaxation/curvature point of view. Even though there is 101... 

Though the sum-over-states formalism of the previous section can be used in semi-empirical treatments, e.g. Pierce [29], it is not appropriate in ab initio calculations for anything but the smallest of chemical systems. This is because of the need for a highly extensive and highly electron-correlated pseudo-spectral series. Faute de mieux, one must turn to less accurate ab initio treatments. For... 

Finite basis set Hartree-Fock calculations yield not only an approximation for the occupied orbitals but also a representation of the spectrum which can be used in the treatment of correlation effects. In particular, the use of finite basis sets facilitates the effective evaluation of the sum-over-states which arise in the many-body perturbation theory of electron correlation effects in atoms and molecules. Basis sets have been developed for low order many-body perturbation theoretic treatments of the correlation problem which yield electron correlation energy components approaching the suh-milliHartree level of accuracy [20,21,22]. 

If high accuracy is required, vibrational effects must be taken into account. In a proper adiabatic Born-Oppenheimer treatment, the groimd state wave function would be written as a product of an electronic and a vibrational wave function. The response of this wave function should then be computed and subsequently used to construct vibronic response functions. The sum-over-states expressions would include contributions from vibrational states in the electronic groimd and excited states. Since each set of vibrational wave functions is tied to a specific electronic state within the adiabatic Bom-Oppenheimer approximation, this approach is not feasible in practice. Hence, the electronic properties are considered as electronic ground state properties and therefore, averaged in a vibrational state of the electronic ground state. 

Two main types of theoretical treatment have been used for the calculation of MCD terms of organic -electron systems (see Spectrum Prediction Spectrum Simulation). Both use perturbation theory to treat the effect of the magnetic field. Most calculations use an ordinary atomic basis set and a sum-over-states expansion, and a few use London orbitals and finite perturbation theory. 

Here, we review an adiabatic approximation for the statistical mechanics of a stiff quantum mechanical system, in which vibrations of the hard coordinates are first treated quantum mechanically, while treating the more slowly evolving soft coordinates and momenta for this purpose as parameters, and in which the constrained free energy obtained by summing over vibrational quantum states is then used as a potential energy in a classical treatment of the soft coordinates and momenta. 

The classical continuum approximation (13.80) becomes questionable for light nuclei, where inertial moments are reduced and quantum rotational spacings proportionally increased, and in this case the quantum sum over angular momentum states may be substituted. Furthermore, the treatment assumes sufficient free volume for unhindered rotations, and is therefore only appropriate at the lower-density conditions of gaseous reactions. 

The angle brackets remind us that these energy terms are quantum-mechanical average values or expectation values each is a functional of the ground-state electron density. Focussing first on the middle term, the one most easily dealt with the nucleus-electron potential energy is the sum over all 2n electrons (as with our treatment of ab initio theory, we will work with a closed-shell molecule which perforce has an even number of electrons) of the potential corresponding to attraction of an electron for all the nuclei A ... 

In the simple treatment of ID tunneling we have considered an electron tunneling from the sample to the tip, however from symmetry arguments an electron could just as easily tunnel from the tip to the sample. This can only happen however if there are unoccupied states available in the sample. We can estimate the tunneling current by summing over the contributions from all states within the energy interval EF1 — EF2 = eV, shown in Fig. 2... 

In the Bishop and Kirtman (BK) perturbation treatment [17-19] two basic additional assumptions are made. First, when K> is an intermediate excited electronic state it is assumed that, under ordinary non-resonant conditions, one may ignore the optical frequency term zr j. /2- -w in the corresponding energy denominator as compared to the electronic excitation energy. Then, after summing over all intermediate states other than K = 0, one is left with die pure vibrational (hyper)polarizability, P". The latter may be expressed compactly in terms of so-called square bracket quantities. Thus,... 

The double adiabatic approach provides a convenient starting point for a detpt theory (2i). The principle modification is the treatment of the FC factors for the overlap of the proton initial and final eigenstates, when the final proton state is characterized by a repulsive surface. The sum over final proton states becomes an integration over a continuum of states, and bound-unbound FC factors need to be evaluated. An approach can be formulated with methods that have been used to discuss bond-breaking electron-transfer reactions (22). If the motion along the repulsive surface for the dissociation can treated classically. 

An appealing feature of this partitioning is that the correlation (dispersion) energy computationally can be assigned exactly to contributions from orbital pairs. If localized molecular orbitals (LMOs) are used in the correlation treatment, this allows a local (group- or fragment-wise) description of dispersion effects [52]. The correlation energy ( [,) for an HF reference state can be written as a sum over occupied orbital pairs ij... 

The accurate TD approach to tetraatomic systems was first applied to the photofragmentation dynamics of H2HF H2 + HF [42, 43] and HOOH OH + OH [44], in which the two diatomic vibrations were frozen but all the other four internal degrees of freedom were treated exactly. These two theoretical studies established the numerical feasibilities for the accurate TD wave-packet treatment for tetraatomic dynamics problems. Within quick succession, accurate quantum dynamics calculations for the tetraatomic reaction AB + CD A + BCD were reported for the benchmark reaction H2 + OH H + H2O [45-47] and its isotopic reactions DH F OH D + H2O, H F DOH [48] and D2 F OH D F DOD [49]. For the H2 F OH reaction, cumulative reaction probabilities have also been computed by calculating the flux directly without summing over individual reaction probabilities [50-52]. Additionally, accurate dynamics calculations for reactions of HO F CO H F CO2 [53], H2 F CN H F HCN [54, 56], and D2 F CN D F DCN [55] have been reported. With the exception of Refs. [50, 56] which use the time-independent iterative approach to calculate cumulative reaction probabilities, the other works all applied the TD wavepacket approach. In addition, the TD calculation for the reverse reaction of H F H2O H2 F OH has been reported [57]. More recently, state-to-state TD calculations have been reported for the H2 F OH reaction [58, 59] and its reverse reaction H F H2O [60]. 

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