Taylor expansion, molecular systems

As stated in the introduction, conceptual DFT is based on a series of reactivity descriptors mostly originating from a functional Taylor expansion of the E = E[N, v(r)] functional. These (<)nE/3NmSv(r)m ) quantities can be considered as response functions quantifying the response of a system for a given perturbation in N and/or v(r). In the case of molecular interactions (leading to a new constellation of covalent bonds or not), the perturbation is caused by the reaction partner. In Scheme 27.1 an overview of the interaction descriptors up to n 2 (for a more complex tabulation and discussion of descriptors up to n 3, see Refs. [11,12]) is given. 

In nonequilibrium steady states, the mean currents crossing the system depend on the nonequilibrium constraints given by the affinities or thermodynamic forces which vanish at equihbrium. Accordingly, the mean currents can be expanded in powers of the affinities around the equilibrium state. Many nonequilibrium processes are in the linear regime studied since Onsager classical work [7]. However, chemical reactions are known to involve the nonlinear regime. This is also the case for nanosystems such as the molecular motors as recently shown [66]. In the nonlinear regime, the mean currents depend on powers of the affinities so that it is necessary to consider the full Taylor expansion of the currents on the affinities ... 

It is possible to generalize this discussion in a useful way. Spectral measurements invariably assess how a molecular system changes in energy in response to some sort of external perturbation. The example presently under discussion involves application of an external electric field. If we write the energy as a Taylor expansion in some generalized vector perturbation X, we have... 

Let us consider the electronic energy E (x) as a function of some external parameter x. When a molecular electronic system is perturbed in some manner, its total electronic energy changes and may be expressed in terms of a Taylor expansion as... 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

The method FUERZA [169] was developed in order to avoid this ambiguity. In this method, the force constants are defined from a tensor-based formalism as follows. For a N-atom molecule or system, the 3N components of the reaction force JF due to a displacement of the N atoms of the molecular system can be expressed exactly to second order in a Taylor series expansion as... 

These quantities determine the associated quadratic Taylor expansion of the system electronic energy, around the ground-state reference system value, e.g., for the molecular or reactive system as a whole ... 

Prior to applications to SFE, WDAs were originally proposed to give a more realistic description of molecular packing effects in interfacial systems, and Evans [14] has given an interesting discussion of their relative suitability in the context of interfaces and SFE. WDAs are nonperturbative in character and so are not subject to the criticism leveled at the functional Taylor expansion approach. 

The solution for the MEP, as shown above, is time independent. Of course, molecular systems exist in a time dependent universe and they are constantly exercising motions on the PES different than the specific motion described by the MEP. An alternative picture for the reaction path is to allow the nuclei to move on the PES according to Newton s equations of motion. In terms of time, the coordinates of the system at time t, x, can be given by the Taylor expansion... 

The fully relativistic (four-component) LCAO calculations of molecular systems use contracted Gaussian-type spinors as the basis two scalar wavefunctions within a two-component basis spinor are multiplied by a common expansion coefficient, for dimensions n of both the large and small components the total number of variational parameters (the scalar expansion coefficients) is equal to 2n [496]. In the relativistic correlated calculations the atomic basis sets should be optimized in the atomic correlated calculations. As Almlof and Taylor showed [538], atomic basis sets optimized to describe correlations in atoms also describe correlation effects in molecules very well. The two main types of basis sets are used in correlation calculations of molecules basis of atomic natural orbitals (ANO) suggested by Ahnlof and Taylor [538] correlation-consistent (CC) basis set suggested by Dunning [462]. 

However, for the usual molecular case, the external field will influence the internal field. Even within the Born-Oppenheimer approximation, E will distort the electron distribution, thereby modifying the interactions in E. We can take this into account by expressing the energy of the system as a function of the field. We use the Taylor expansion... 

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