Kinetic energy noninteracting

Thus the interacting multi-electron system can be simulated by the noninteracting electrons under the influence of the effective potential l eff(r)- Kohn and Sham [51] took advantage of the fact that the case of non-interacting electrons allows an exact computation of the particle density and kinetic energy as... 

Exc accounts for the energy of exchange interactions, correlation effects, and the difference between the exact kinetic energy and that of the reference system of noninteracting electrons with the density p(r). 

The third term on the right hand side of this expression is the single-particle kinetic energy of the noninteracting electrons whereas the functional Exc[p contains the additional contribution to the energy that is needed to make Eq. (8) equal to Eq. (6). 

One now replaces the interacting T[p in Equation 4.34 by the noninteracting 7 s[pJ. This means that any kinetic energy missing as a result of this replacement must be included in Exctpl clearly, T[p > Ts[p]. Equation 4.34 now becomes... 

The Kohn-Sham equations can be obtained from the minimalization of the noninteracting kinetic energy after expressing it with one-electron orbitals. Because Tl 0 is generally a linear combination of several Slater determinants, the form of the... 

Kohn-Sham equations is rather complicated for an arbitrarily selected set of weighting factors, and has to be derived separately for every different case of interest. For a spherically symmetric external potential and equal weighting factors, however, the Kohn-Sham equations have a very simple form, as shown in Ref. [72], In this case the noninteracting kinetic energy is given by... 

It is also possible to extend the Kohn-Sham formalism by defining an energy term Ts that includes the kinetic energy of the noninteracting system, and the total... 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

The reason for calling I [n] the noninteracting kinetic energy functional is recognized immediately when the GS problem of a system of N noninteracting electrons, moving in an external potential rs(r), is considered. In analogy with Eq. (9) we write... 

The exchange-correlation (xc) energy functional defined above is shown to consist of two contributions a difference between the interacting and noninteracting kinetic energy functionals and the nonclassical part of the electron-electron interaction energy functional. Using Eq. (44) we rewrite Eq. (14) as... 

The HF GS density nGs(f), occurring in Eq. (64), coincides with the density calculated according to Eq. (29) from [the solutions of the HF equations (33)], because the two ways of calculation of s at equivalent. By adding and subtracting the noninteracting kinetic energy functional to HF functional in... 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r) as a sum over densities of the occupied orbitals, that is, p(r) = Xl<)>,l2, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential v(r) of the nuclei and the classical Coulomb repulsion VCoul(r) within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been folded into a local potential vxc r) ... 

To utilize Eq. 7.11, Kohn and Sham introduced the idea of a fictitious reference system of noninteracting electrons which give exactly the same electron density distribution as the real system has. Addressing electronic kinetic energy, let us define the quantity A(T[p ] ) (don t confuse Greek delta A, an increment, with the differential operator del V) as the deviation of the real electronic kinetic energy from that of the reference system ... 

The integrals to be summed are readily calculated. Note that DFT per se does not involve wavefunctions, and the Kohn-Sham approach to DFT uses orbitals only as a kind of subterfuge to calculate the noninteracting-system kinetic energy and the electron density function see below. 

The locality hypothesis can be tested in a noninteracting model, in which the functional Fs is replaced by 7 . The kinetic energy orbital functional is T = ni 0 If 10... 

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