Natural spin-orbital functional

Cioslowski J, Pemal K (1999) Constraints upon natural spin orbital functionals imposed by properties of a homogeneous electron gas, J. Chem. Phys, 111 3396-3400... 

The results reported in Table V were calculated by means of a basis consisting of four s functions, three p functions, two d functions, and one / function of the type of Eq. III.57 for rj — 2.2. The corresponding 20-term function had an energy of —2.901231 at.u.He, and, by going over to natural spin orbitals, one obtains a 10-term function with the same energy. 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

M. Levy, Universal variational functionals of electron-densities, Ist-order density-matrices, and natural spin-orbitals and solution of the v-representability problem. Pmc. Natl. Acad. Sci. U.S.A. 76(12), 6062-6065 (1979). 

For a multi-Slater determinant wave function, orbitals which satisfy Eq. (3.6), and therefore Eq. (3.7), can still be defined. For these orbitals, referred to as the natural spin orbitals, the coefficients nt are not necessarily integers, but have the boundaries 0 n, 1. 

From a strictly quantum-mechanical point of view, the question of valence-shell expansion and d-orbital participation has two different aspects. One may either look for d-character in Lowdin s natural spin-orbitals of the complete, but unknown, total wave-function or one may ask the question whether the agreement with experimental results is ameliorated significantly when d-orbitals are included in the basis functions of approximate M. 0. calculations. It is a well known tendency for approximate calculations always to ameliorate in certain aspects when the basis is expanded, but also that the extent of this amelioration does not always bear a direct relation to the final results of far more sophisticated calculations. 

The natural spin-orbitals, which were first introduced by Lowdin179 in 1955, were shown to produce the most rapidly convergent expansion of the first-order density matrix,180 Thus, the number of determinants needed for any required degree of accuracy in the wavefunction can be greatly reduced if the determinantal wave-functions are constructed from natural orbitals at each stage in the calculation. 

A good first approach to a quantum mechanical system is often to consider one-electron functions only, associating one such function, a spin-orbital , with one electron. Most popular are the one-electron functions which minimize the energy in the sense of Hartree-Fock theory. Alternatively one can start from a post-HF wave function and consider the strongly occupied natural spin orbitals (i.e. the eigenfunctions of the one-particle density matrix with occupation numbers close to 1) as the best one-electron functions. Another possibility is to use the Kohn-Sham orbitals, although their physical meaning is not so clear. 

The natural spin-orbitals a,(l) corresponding to a many-electron wave function are defined to be the set in terms of which the first-order density matrix is diagonal [36-52], Thus, if we solve the secular equation... 

We normalize Pci such that the coefficient of is equal to 1. The spin orbitals (pa, natural spin orbitals of the respective lEPA pair correction function My 22) jhis will be the case for all of the methods considered and we shall not stress this again. One ought to add the abbreviation PNO (for Pair-Natural-Orbitals, sometimes interpreted as Pseudo-Natural-Orbitals) to the terms lEPA and CEPA as well, i.e. to speak of lEPA-PNO and CEPA-PNO rather than just of lEPA and CEPA. Using the PNO s one gets mutually orthogonal 0if though the PNO s of different pairs are nonorthogonal. Also the explicit expressions for Hab are not too complicated. 

The single concept, which is defined and discussed in these articles, that without doubt has had the most penetrating impact on the whole field of quantum chemistry, is the concept of the natural spin orbitals. This concept is, in principle, very simple the set of MOs that makes the first-order density matrix diagonal. Lowdin starts out by defining a hierarchy of reduced density matrices for a general wave function ... 

In analogy to the HF result of Eq. (4.39), A is called the occupation number of the natural spin orbital rji in the wave function O. 

The function Xi is called a natural spin orbital (NSO). The eigenvalue v, is the occupation number of Xi- It may be shown that the sum of occupation numbers is equal to N. There is generally an infinite set of NSOs, except when the wave function is approximated by a finite number of Slater determinants. 

W. Kohn and L. H. Sham, Phys. Rev. A, 140, 1133 (1965). Self-Consistent Equations Including Exchange and Correlation Effects. M. Levy, Proc. Natl. Acad. Sci. U.S.A., 76, 6062 (1979). Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. R. G. Parr and W. Yang, Eds., Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1988. J. Labanowski and J. Andzelm, Eds., Density Functional Methods in Chemistry, Springer Verlag, Heidelberg, 1991. L.J. Bartolotti and K. Flurchick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 187-216. An Introduction to Density Functional Theory. 

LOwdin, P.-O. (1955a). Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 97, 1474-1489 ibid. (1955b). Quantum theory of many-particle systems. II. Study of the ordinary hartree-fock approximation. Phys. Rev. 97, 1490-1508 ibid. (1955). Quantum theory of many-particle systems, in. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects. Phys. Rev. 97,1509-1520 ibid. (1960). Expansion theorems for the total wave function and extended Hartiee-Fock schemes. Rev. Mod. Phys. 32, 328-334. 

Almost all contemporary electronic structure calculations involve the use of one-electron functions called spin orbitals. The calculations usually commence with the computation of canonical Hartree-Fock (spin) orbitals ( molecular orbitals or MOs) which are eigenfunctions of the Fock operator. In turn, MOs are input into complicated functionals that yield approximate values of the correlation energy. Formal differentiation of those functionals with respect to strengths of external perturbations results in formulae for effective one-electron density matrices F(r, r ) 3 -t33 from which any one-electron, first-order response property can be calculated. Finally, diagonalization of F(r, i ) produces natural (spin) orbitals (NOs) (see Natural Orbitals). 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

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