The Perturbation Formalism

Let us start by considering an. /V-particle reference system described by the Hamiltonian. (x, p , j, which is a function of 3N Cartesian coordinates, x, and their conjugated momenta p,. We are interested in calculating the free energy difference between this system and the target system characterized by the Hamiltonian 

The difference in the Helmholtz free energy between the target and the reference systems, AA, can be written in terms of the ratio of the corresponding partition 

As has already been discussed in the context of (1.19), the probability density function of finding the reference system in a state defined by positions x and momenta px is 

( )0 denotes an ensemble average over configurations sampled from the reference state. This is the fundamental FEP formula, which is the basis for all further developments in this chapter. It states that AA can be estimated by sampling only equilibrium configurations of the reference state. 

Note that integration over the kinetic term in the partition function, (2.3), can be carried out analytically. This term is identical for the solvated and the gas-phase molecule in the first example given at the beginning of this section. Thus, it cancels out in (2.2), and (2.7) becomes 

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The formulas for free energy differences, (2.8) and (2.9), are formally exact for any perturbation. This does not mean, however, that they can always be successfully applied. To appreciate the practical limits of the perturbation formalism, we return to the expressions (2.6) and (2.8). Since AA is calculated as the average over a quantity that depends only on A17, this average can be taken over the probability distribution Po(AU) instead of Pq(x. p ) [6], Then, AA in (2.6) can be expressed as a onedimensional integral over energy difference... 

By taking Eq. (1.33) as the starting unperturbed Eq. (1.2), one can analyze delocalization corrections to the localized Lewis structure picture by the perturbative formalism of Eqs. (1.3)-(1.5) and Section 1.4. Valency and bonding phenomena can thereby be dissected into localized and delocalized contributions in a numerically explicit manner. This, in overview, is the strategy to be employed for chemical phenomena throughout this book. 

In order to properly treat electrode effects, it is essential to use the perturbation formalism. The small-load approximation gives the wrong results. The situation is particularly dangerous in dry environments. In liquids, the shortcomings of the small-load approximation are less severe. 

In Section II we give a detailed account of the perturbation formalism of Bloch and de Dominicis, which for our particular purpose is extended to the case of particles moving in an external potential. Two kinds of expansion of the grand partition function are considered the first one in powers of the interaction strength A, the second one in powers of the chemical activity. 

In order to apply the perturbation formalism, we now need to calculate the matrix elements of v and v with respect to the unperturbed single-particle wave functions. These will be taken as the eigenfunctions of the momentum operator, normalized in a cubic box of volume Q with periodic boundary conditions ... 

Let us now come to some of the more recent implementations of the perturbation formalism into computer codes. The first implementation of the contributions at DFT level by Malkin et al. [65,66] included by FPT (neglecting H ) and evaluated the and H contributions by finite difference of second-order expressions for the remaining perturbations, based on the Kohn-Sham orbitals spin-polarized by the FC term. While the initial implementation [66] included only and thus evaluated, the method... 

Simulations of femtosecond time-resolved photoelectron spectra have first been performed with this formalism for a three-mode model of the S1-S2 conical intersection in pyrazine (see below). With the same technique, Schon and Koppel have explored the real-time detection of pseudo-rotational motion due to Jahn-Teller and pseudo-Jahn-Teller couplings in Na3. The same problem has been addressed within the perturbational formalism by Dobbyn and Hutson.Charron and Suzor-Weiner have treated the femtosecond photoionization dynamics of Nal within the non-perturbative approach, considering also passive control scenarios to influence the fragmentation pattern of Nal. ... 

In the Brillouin-Wigner perturbation formalism, the following identity is used... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

The indicated formal potential E° n of the corresponding monomer (-1.17V) in solution is very near that of the surface film (-1.13V vs. SSCE). That formal potentials of surface films on chemically modified electrodes are near those of their corresponding dissolved monomers (13,18) is actually a common, and quite useful, observation. In the present case, it demonstrates that the electronic structures of the porphyrin rings embedded in the polymer film are not seriously perturbed from that of the monomer. 

In this section, we discuss applications of the FEP formalism to two systems and examine the validity of the second-order perturbation approximation in these cases. Although both systems are very simple, they are prototypes for many other systems encountered in chemical and biological applications. Furthermore, the results obtained in these examples provide a connection between molecular-level simulations and approximate theories, especially those based on a dielectric continuum representation of the solvent. 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

To apply the perturbation-theory formalism we can first separate H into diagonal (unperturbed) and off-diagonal (perturbation) matrices,... 

Although any of these decompositions might be employed in the formal machinery of perturbation theory, one can expect that choices of F(0) for which the perturbation elements in (Per9 are small will lead to more rapid convergence, and thus serve as better models. 

E. Schrodinger, Ann. Phys. 80 (1926), 437. The quantal formalism substantially follows the classical method developed by Lord Rayleigh (Theory of Sound [1894]) and is commonly referred to as Rayleigh-Schrodinger perturbation theory. ... 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

In this section, we extend the above formalism to that for an alloy surface within the CPA, which serves as the model for the pre-chemisorption substrate. The model discussed here is based on that of Ueba and Ichimura (1979a,b) and Parent et al (1980). For a comprehensive introduction to alloy surfaces see Turek et al (1996). A feature of surface-alloy models, which is different from bulk ones, is that the CP in layers near the surface is different from that in the bulk, due to the surface perturbation. Moreover, the alloy concentration in the surface layers may be quite different from that in the bulk, a feature known as surface segregation. (See Ducastelle et al 1990 and Modrak 1995 for recent reviews.) We assume that both of these surface effects are confined to the first surface layer only. 

This chapter aims to present the fundamental formal and exact relations between polarizabilities and other DFT descriptors and is organized as follows. For pedagogical reasons, we present first the polarizability responses for simple models in Section 24.2. In particular, we introduce a new concept the dipole atomic hardnesses (Equation 24.20). The relationship between polarizability and chemical reactivity is described in Section 24.3. In this section, we clarify the relationship between the different Fukui functions and the polarizabilities, we introduce new concepts as, for instance, the polarization Fukui function, and the interacting Fukui function and their corresponding hardnesses. The formulation of the local softness for a fragment in a molecule and its relation to polarization is also reviewed in detail. Generalization of the polarizability and chemical responses to an arbitrary perturbation order is summarized in Section 24.4. 

The Perturbation Theory Formalism. The quantum mechanical model employed here is the conventional time-dependent pertur-... 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

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