Ground-state electron density electronic systems

Before proceeding further, it is necessary that we choose a method for constructing the internal potential, w(r). The insight of Kohn and Sham was to choose the internal potential so that the systems defined by the true Hamiltonian [Eq. (11)] and the model Hamiltonian [Eq. (37)] have the same ground-state electron density [15], This can occur only if the ground-state energy density functionals, Ev p [cf. Eq. (21)], and... 

Once in game the external applied potential provides the second Hohenberg-Kohn (HK2) theorem, hr short, HK2 theorem says that the external applied potential is determirred up to an additive constant by the electronic density of the iV-electrorric system ground state . In mathematical terms, the theorem assures the validity of the variatiorral principle applied to the density functional (4.381) relation, i.e., (Emzerhof, 1994)... 

The minimum value in EKs p(r)] corresponds to the exact ground-state dectron density. To determine the actual energy, variations in E p(r)] must be optimized with respect to variations in p, subject to the orthonormality constraints. In KS-DFT, an arriiidal reference system of noninteracting dectrons is constracted, which has exactly the same electron density as the real molecular system. Therefore, from a computational viewpoint, the KS version of DFT leads to a mean-field solution, which is only an approximation. 

A second approach, on which most modern band calculations are based, re tg on the density-functional theory of Hohenberg, Kohn and Sham. This theory established that all properties of an interacting electron system may be considered as functionals of the ground-state charge density p(r) rather than of an applied external potential approximate grougd-state energy func-... 

The degenerate MOs 02 and 03 should have one nodal plane each, and these should be perpendicular to each other.If we take one plane as shown in (VIII), we can immediately write down 02. The node for 03 is given in (IX). It is obvious that X6, XI, X2 have coefficients of the same sign, and that C2 = ce = —cs = —cs, and also c = —C4. However, c need not equal C2 as these atoms are differently placed with respect to the nodal plane. To determine these coefficients, we will use the fact that the neutral ground-state tt densities are all unity in this system. We consider first atom number 1. Its electron density due to two electrons in 0i and two electrons in 02 is... 

The so-called Flohenberg-Kolm [ ] theorem states that the ground-state electron density p(r) describing an A-electron system uniquely detemiines tlie potential V(r) in the Flamiltonian... 

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

Configuration interaction, which is necessary in treatments of excited states and desirable in calculations of spin densities, is more complex with open-shell systems. This is because more types of configurations are formed by one-electron promotions. These configurations (Figure 5) are designated as A, B, Cq, C(3 G is the symbol for a ground state. Configurations C and Cp have the same orbital part but differ in the spin functions. 

These limitations, most urgently felt in solid state theory, have stimulated the search for alternative approaches to the many-body problem of an interacting electron system as found in solids, surfaces, interfaces, and molecular systems. Today, local density functional (LDF) theory (3-4) and its generalization to spin polarized systems (5-6) are known to provide accurate descriptions of the electronic and magnetic structures as well as other ground state properties such as bond distances and force constants in bulk solids and surfaces. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

Instead of treating all electrons in the metal plus adsorbate system individually, one considers the electron density of the system. Hohenberg and Kohn (Kohn received the 1999 Nobel Prize in Chemistry for his work in this field) showed that the ground state Eq of a system is a unique functional of the electron density in its ground state Wq- Neglecting electron spin, the energy functional can be written as... 

The ground state of the system corresponds to 2N electrons occupying the lowest-energy Ej levels, so that the electron numeral density n r) is... 

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