Parr theorem

The Parr theorem, regarded as the finite difference formula associated with the Hellmann-Feynman theorem, has some interesting consequences. In most quantum-chemical models, the difference of two Hamiltonians on the right-hand side is a one-electron operator. Consequently, at least in principle, in the calculation of the conformational energy difference the evaluation of complicated two-electron... 

The DFT approach (for an excellent introduction, see Parr and Yang 1989) is different and somewhat simpler. The electron density p(r) has been recognized to be a feature that uniquely determines all properties of the electronic ground state (1st Hohenberg-Kohn theorem). Instead of minimizing E with respect to coefficients of the wave function as in HF, E is minimized with respect to the electron density p... 

Applying Gauss theorem, leading to the Politzer-Parr core-valence separation in atoms [61]... 

Density functional theory (DFT) is based on the Hohenberg-Kohn theorem, which states that there is a functional which gives the exact ground-state energy for the exact electron density. DFT models have become very popular because they are not more costly than Hartree-Fock models. The reason is that in the HF, Cl and MP models, a wavefunction for an N-electron system depends on 3N coordinates, whereas in the DFT approach, the electron density depends on only three coordinates, irrespective of the number of electrons. The problem is that the exact functional would be the Schrodinger equation itself Several approximate functionals have been developed by many authors (Becke, Parr, Perdew, and others) and different forms of the functional can yield slightly different results. Some of the most common DFT models are ... 

The linear acenes, benzene to pentacene, are used as examples of the CURES-EC procedure. The results obtained utilizing MINDO/3 and AMI are compared. In addition to calculating the Ea by subtracting the energies of the optimized form, the LUMO of the neutral is compared with the experimental Ea. The electron affinity of hexacene has been estimated from the electronegativity and experimental ionization potential. As a further example of the use of CURES-EC, both the ionization potential and electron affinity of heptacene are estimated. The Ea of octacene and nova-cene are calculated for comparison to values obtained by using Koopman s theorem and a semi-empirical method based on a variable-parameter modification of the Pariser Parr Pople (PPP) approximation to the Hartree Fock equation [10]. 

A theorem by Parr can be regarded as an integrated form of the Hellmann-Feynman theorem ... 

The writer (March 1981) has emphasized that an alternative route to Eq. (Al.l) for calculating x(Z,N) is via Feynman s theorem, which yields (see March, 1975 March and Parr, 1980)... 

Huckel theory for the even alternant hydrocarbons leads to the Coulson-Rushbrooke theorem and some other characteristic results shown by McLach-lan to be valid also in the Pariser-Parr-Pople model. These are the well-known pairing relations between electronic states of alternant hydrocarbon cat-and anions. This particle-hole symmetry is analogous to the situation discussed in Chapter 4 for electrons and holes in atomic subshells. 

Hohenberg and Kohn proved their theorems only for nondegenerate ground states. Subsequently, Levy proved the theorems for degenerate ground states (see Parr and Yang, Sec. 3.4). 

The major consequence of this theorem consists in defining of the total energy of a system as a function of the electronic density function in what is known as the density functional (Parr Young, 1989 Putz, 2003) ... 

March, N. H. Parr, R. G. Chemical-potential, Teller s theorem, and the scaling of atomic and molecular energies. Pmc. Natl. Acad. Sci. 1980, 77, 6285-6288. 

Virial theorem has an important role in quantum mechanics. It proved to be also very useful in density functional theory (see e.g. Parr and Yang, 1989). Several forms of virial theorem have been proposed in ground-state density functional theory, for example, local virial theorem (Nagy and Parr 1990), differential virial theorem (Holas and March 1995), regional virial theorem (Nagy 1992), and spin virial theorem (Nagy 1994b). In this chapter, local and differential virial theorems are extended to excited states in the frame of density functional theory. 

It has the same form as the ground-state local virial theorem (Nagy and Parr 1990). 

Modern DFT is founded on the Hohenberg-Kohn theorems and the Kohn-Sham equations. These are presented in detail in textbooks as well as in the review literature on the subject (Parr and Yang 1989, Koch and Holthausen 2001, Eschrig 1996, Gross and Kurth 1994, Salahub et al. 1994). However, to set the stage for a discussion of the relativistic case, a brief summary of the nonrelativistic foundations serves as a convenient starting point. 

Then, while considering the finite-difference approximations to the electronegativity and chemical hardness definitions, in a Koopmans theorem environment (Koopmans, 1934) they take the woiking forms (Parr Yang, 1989 Chermette, 1999 Geerlings et al., 2003 Putz, 2012b) ... 

This first theorem by Hohenberg and Kohn (HK-I) is not difficult to prove (Parr and Yang 1989), but for a chemist, the essentials of HK-I that given a density, only one external potential corresponds to that density, are intuitively clear. A pictorial representation of HK-I is shown... 

The VAEs of bases were also predicted using theoretical methods such as semiempirical Pariser-Parr-Pople (PPP) (Gompton et al. 1980 Younkin et al. 1976) and ab initio methods (Sevilla et al. 1995). Since these anion states are metastable with respect to the autodetachment of the electron, special care was taken in the aforementioned theoretical approaches to handle these anion states by employing confined basis set, Koopmans theorem approximation, and empirical determination of parameters to scale the theoretical results to the experimentally measured anion state energies for other compounds. Sevilla et al. (1995) used calculated values for VAEs of benzene, naphthalene, pyridine, pyrimidines, and uracil, which have experimentally known values to scale theoretically calculated VAEs for the DNA bases. A comparison of the experimental VAEs of the bases to those predicted theoretically is presented in O Table 34-1. From O Table 34-1, we see that the calculated VAE values are in excellent agreement with experimental VAEs and the difference between theory and experiment lies in the range... 

---