To keep the Dirac exchange [15] of the electron gas part complementary to Exc of Equation 5.7, the classic form of linear combinations is Fj 1 f , for hybrid functionals. Thus, the number of fitted [Pg.119]

CALCULATION OF THE ELECTRONIC STRUCTURE OF ANTIFERROMAGNETIC CHROMIUM WITH A SINUSOIDAL SPIN DENSITY WAVE BY THE METHOD OF DIRAC FUNCTION LINEAR COMBINATION [Pg.139]

If we were to assume a basis set expansion for the spinor of the type of a linear combination of atomic orbitals (LCAO) we could differentiate the La-grangian functional directly and would obtain equations in matrix form (compare the Dirac-Hartree-Roothaan equations in chapter 10). Here, we proceed in a more general way and proceed with the general method of variations. The variation of any of the matrix elements over an operator o containing in L[ tpi, ey ] may be written as the limit for infinitely small variations of a given orbital ipi as [Pg.307]

The function / is a probability density, indicated by the shading of the lower right part of Fig. 6. To bring about a decomposition of into eigenstates, one must choose the function / as a linear combination of Dirac delta functions (i.e., as an appropriate distribution). [Pg.115]

The RQDO radial, scalar, equation derives from a non-unitary decoupling of Dirac s second order radial equation. The analytical solutions, RQDO orbitals, are linear combinations of the large and small components of Dirac radial function [6,7] [Pg.52]

The fact that the generalized coordinate / is a linear combination of all bath modes and that the potential is quadratic in the bath variables allows one to express the potential of mean force w[f ] in terms of a single quadrature over the system coordinate q. The detailed derivation is presented in Ref. 42, the main technical trick being the usual use of the Fourier representation of the Dirac 8 functions. The resulting expression is [Pg.636]

The set of B-splines of order k on the knot sequence f< forms a complete basis for piecewise polynomials of degree fc — 1 on the interval spanned by the knot sequence. We represent the solution to the radial Dirac equation as a linear combination of these B-splines and work with the B-spline representation of the wave functions rather than the wave functions themselves. [Pg.142]

In the first chapter, we saw that if we wanted to rotate the 2px function, we automatically also needed its companion 2py function. If this is extended to out-of-plane rotations, the 2/ function will also be needed. The set of the three p-orbitals forms a prime example of what is called a linear vector space. In general, this is a space that consists of components that can be combined linearly using real or complex numbers as coefficients. An n-dimensional linear vector space consists of a set of n vectors that are linearly independent. The components or basis vectors will be denoted as fi, with I ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as / >, which characterizes them as so-called kef-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called fera-functions, denoted as /t I The scalar product of a bra and a ket yields a number. It is denoted as the bracket fk fi). In other words, when a bra collides with a ket on its right, it yields a scalar number. A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given. [Pg.12]

In our discussion, we have so far examined the electron density in the sjrin-orbital and orbital spaces. Let us now consider the electron density in ordinary space. Of particular interest are the expectation values of operators that probe the presence of electrons at particular points in space. Thus, the one-electron first-quantization operator in the form of a linear combination of Dirac delta functions [Pg.64]

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