Exact kinetic energy

Anotiier way of justifying the use of eq. (6.4) for calculating tire kinetic energy is by reference to natural orbitals (eigenvectors of the density matrix. Section 9.5). The exact kinetic energy can be calculated from the natural orbitals (NO) arising from tire exact density matrix. 

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). 

For particulate radiation or any very rapidly moving mass, the expression previously given for the kinetic energy. Jntr2. is not accurate when the velocity approaches that of the velocity of light. The theory of relativity requires a correction be made, and the exact kinetic energy. /. may be calculated in terms of the mass. iiiu. of light in vacuum, c. as follows ... 

The first term, the kinetic energy, is difficult to calculate directly from the density, and it is for this reason that the molecular orbitals mentioned above are introduced a very good approximation to the kinetic energy corresponding to the density can be calculated from the orbitals as it would be in HF theory. This approach does not however yield the exact kinetic energy because it assumes that the electrons in each orbital do not interact with electrons in other orbitals. The exact exchange-correlation functional must therefore contain a corrective term to incorporate the effect of electronic interactions on their kinetic energy. In practice, such a term is not explicitly included in common functionals. 

The evaluation of the classical thermodynamic energy I4, is performed by including the exact kinetic energy fn/cT and the potential energy averaged over the distribution given in Eq. (4.20). We then obtain... 

J. Pesonen, Exact kinetic energy operators for polyatomic molecules, in Applications of Geometric Algebra in Computer Science and Engineering, L. Dorst, C. Doran, and J. Lasenby, eds., Birkhauser, Boston, 2002, p. 261-270. 

The exact crystal Hamiltonian H is given by Eq. (23) and H0 is of the form given by Eq. (47) the force constants Fp f> are not given by Eq. (48), however, but they are chosen such as to minimize Avar- Neglecting the difference between the exact kinetic energy operators [(25) and (26)] and their harmonic approximations (see Section III,A), one obtains... 

GEA was tested. But the case of a nAS-representable pair of electron densities is much more general. It is striking to note that, the second-order GEA contribution to T ad pA, Pb is non-positive for all pairs of electron densities (for uAB-representable and not uAB-representable pairs alike). This follows from its explicit analytic form given here in the second line of Eq. 71. We recall now that the second-order GEA term (T2 in Eq. 69) is closely related to the von Weizsacker functional T [p 75 (T [p = 9T2[p]), which is the exact kinetic energy functional for one- and two-spin-compensated electron systems. Using Tsw p] in approximating T ad pA, Pb]... 

Let us have a look at the exact kinetic energy. In general, it is written as... 

The main difference and the potential of this approach lies in the detail that Vxc(r) includes not only the exchange in the Hartree-Fock (HF) equations, but also the correlation (referred to all that is missed by the Hartree-Fock approach) components. In addition, the difference between the exact kinetic energy of the system and the one calculated from the KS orbitals are included. This method states that Vxc(r) is the best way to describe the fact that every electron aims to maximize the attraction from the nuclei and to minimize the repulsion from the rest of the electrons along its constant movement within an entity (atom or molecule). Vxc(r) describes the exchange correlation... 

It can be easily noted, however, that Exc does not correspond to the Ec + Ex of the standard HF based definitions, nor to the definition of Ex used in the exact solutions of the Schrodinger equation which are not based on an MO theory. The main difference is that Exc contains a portion of the exact kinetic energy. 

How closely Trsn n approaches the exact kinetic energy for a noninteracting system depends on our choice of the primitive orbital set. In the case reviewed here [85], the transformed orbitals belong to the Clementi-Roetti-type set. With this choice we obtain l " An = 14.593163 hartrees. 

The external potential operator Vext is equal to Vne for A = 1, but for intermediate A values it is assumed that Vext(A) is adjusted such that the same density is obtained for A = 1 (the real system), for A = 0 (a hypothetical system with non-interacting electrons) and for all intermediates A values. For the A = 0 case, the electrons are noninteracting, and the exact solution to the Schrodinger equation is given as a Slater determinant composed of (molecular) orbitals, and the exact kinetic energy functional is given in eq. (6.6). 

The term [VrP F( )] /p vi ) is known as the von Weizsacker kinetic energy fw[P i (r)] [4]. Hence, in the other words, Corollary 1.1 tells that square-integrable. Usually, the von Weizsacker term is only a part of the total many-electron kinetic energy [4]. The exception is the Hartree-Fock 2-electron model systems for which fw[p>i (r)] is the exact kinetic energy. We further have... 

The exact kinetic energy expression for a polyatomic molecule in a space-fixed coordinate system (SFCS cf. Appendix I available at booksite.elsevier.com/978-0-444-59436-5) bas been given in Chapter 6 (Eq. 6.39). After separation of the center-of-mass motion, the Hamiltonian is equal to H = T + V. where V represents the electronic energy playing the role of the potential energy for the motion of the nuclei (an analog of (/ ) from Eq. (6.8), and we assume the Bom-Oppenheimer approximation). In the hyperspherical democratic coordinates, we obtain ... 

The exchange-correlation term Exc is defined as the exact kinetic energy T p minus the kinetic energy in the KS representation and the exact... 

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