Kohn Formulation

We introduce the 5—matrix version of the Kohn variational principle (KVP) for a reactive system with chemical arrangements labeled by r. For each arrangement we have a radial scattering coordinate Rr and internal coordinates q. A simple way to start is by writing a reactive 5—matrix element in the distorted Bom representation  

Using this trial wavefunction in the Kohn variational expression yields the following stationary 5--matrix element  

We note that the incoming wave boundary conditions in the bra state in Eqs. (2.2) and (2.4) can be enforced by not complex conjugating radial functions in bra states of Eq. (2.7). This inner product is called the biorthogonal inner product [29], and is formally related to the use of complex scaled coordinates and absorbing boundary conditions. 

As we have shown, the KVP reduces the quantum scattering problem to choosing basis functions, computing matrix elements of the Hamiltonian, and performing a linear algebra calctdation to obtain M Mq. Before proceeding with the discussion of sensitivity analysis, we make some qualitative remarks concerning the KVP. 

The KVP resembles the Rayleigh-Ritz variational principle [30] (RRVP). Indeed, if the scattering variational formulation were applied to a bound state problem, for which = 0 = Sji r nr E) the KVP reduces to 

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But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

In the Hohenberg-Kohn formulation, the problem of the functional iV-representability has not been adequately treated, as it has been assumed that the 2-matrix IV-representability condition in density matrix theory only implies an N-representability condition on the one-particle density [21]. Because the latter can be trivially imposed [26, 27], the real problem has been effectively avoided. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

The developments in the implementation of the Kohn-Sham approach, helped also to formulate alternative formalisms aiming at approaching the linear scaling195 218"227. In particular, the divide-and-conquer approach of Yang218 have attracted much attention. 

Lately, the CP-MD approach has been combined with a mixed QM/MM scheme [10-12] which enables the treatment of chemical reactions in biological systems comprising tens of thousands of atoms [11, 26]. Furthermore, CP-MD and mixed QM/MM CP-MD simulations have also been extended to the treatment of excited states within a restricted open-shell Kohn-Sham approach [16, 17, 27] or within a linear response formulation of TDDFT [16, 18], enabling the study of biological photoreceptors [28] and the in situ design of optimal fluorescence probes with tailored optical properties [32]. Among the latest extensions of this method are also the calculation of NMR chemical shifts [14]. 

The development of the method started in the mid 1920 s with the work of Thomas and Fermi [8, 9]. The aim was to formulate an electronic structure theory for the solid state, based on the properties of a homogeneous electron gas, to which we introduce a set of external potentials (i.e. the atomic nuclei). The original formulation, with later additions by Dirac [10] and Slater [11], was, however, inadequate for accurate description of atomic and molecular properties, and it was not until the ground-breaking work of Kohn and coworkers in the mid 1960 s that the theory was put in a form more suited to computational chemistry [12,... 

Whereas the classic Kohn-Sham (KS) formulation of DFT is restricted to the time-independent case, the formalism of TD-DFT generalizes KS theory to include the case of a time-dependent, local external potential w(t) [27]. 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

The conclusion that it may be possible to formulate the quantum mechanics of many-electron systems solely in terms of the single-particle density was put on a firm foundation by the two Hohenberg-Kohn theorems (1964), which are stated below, without proof. 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

Similar relations can be obtained for the nonlinear/ functions. Kohn-Sham orbital formulations of these nonlinear responses can be constructed along the lines described previously [32] and will be presented elsewhere. 

Higher-order derivatives with respect to external potential define xi(r, r1), Xi(r, r1, r"), etc., and their response with N define j(r, rJ), g2(r, r, / ), etc. This chain of derivatives is diagrammatically depicted in Figure 25.1 [22]. Thus, an exact one-electron formulation of all chemical responses (linear and nonlinear hardness, FF) in terms of Kohn-Sham orbital of the unperturbed system was derived [22b]. 

Alternative Formulation of the Kohn-Sham Approach Generalized... 

Nonetheless, Eq. (95) is perhaps the most natural generalization of the Kohn-Sham formulation to g-density functional theory. Indeed, Ziesche s first papers on 2-density functional theory feature an algorithm based on Eq. (95), although he did not write his equations in the potential functional formulation [1, 4]. The early work of Gonis and co-workers [68, 69] is also of this form. 

The second approach to this problem is to derive orbital-based reformulations of existing algorithms based on the spatial representation of the g-density. The resulting formulations are in the spirit of the orbital-resolved Kohn-Sham approach to density functional theory. 

Dispersion interactions are, roughly speaking, associated with interacting electrons that are well separated spatially. DFT also has a systematic difficulty that results from an unphysical interaction of an electron with itself. To understand the origin of the self-interaction error, it is useful to look at the Kohn-Sham equations. In the KS formulation, energy is calculated by solving a series of one-electron equations of the form... 

In the next section we shall recall the definitions of the chemical concepts relevant to this paper in the framework of DFT. In Section 3 we briefly review Strutinsky s averaging procedure and its formulation in the extended Kohn-Sham (EKS) scheme. The following section is devoted to the presentation and discussion of our results for the residual, shell-structure part of the ionization potential, electron affinity, electronegativity, and chemical hardness for the series of atoms from B to Ca. The last section will present some conclusions. 

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