Download Hyperspherical Harmonics and Generalized Sturmians by John S. Avery PDF

By John S. Avery

This ebook explores the connections among the idea of hyperspherical harmonics, momentum-space quantum concept, and generalized Sturmian foundation capabilities; and it introduces tools that could be used to resolve many-particle difficulties at once, with out using the self-consistent-field approximation. the tactic of many-electron Sturmians deals an engaging and clean substitute to the standard SCF-CI tools for calculating atomic and molecular constitution. while many-electron Sturmians are used, and while the foundation power is selected to be the horny power of the nuclei within the approach, the subsequent benefits are provided: the matrix illustration of the nuclear appeal capability is diagonal; the kinetic strength time period vanishes from the secular equation; the Slater exponents of the atomic orbitals are immediately optimized; convergence is fast; a correlated strategy to the many-electron challenge could be acquired without delay, with no using the SCF approximation; and excited states should be got with strong accuracy. viewers: The e-book might be of curiosity to complex scholars and learn employees in theoretical chemistry, physics and arithmetic.

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Sample text

1 are those which are often used t o label the direct-space paraboloidal orbitals. 114) where and where is a Jacobi polynomial. 112). For more details concerning these relationships, the interested reader is referred t o the Russian literature and t o the papers of Aquilanti and his coworkers. 1 Decomposition of monomials into harmonic polynomials The indices i j k are assumed to be all unequal. The terms in brackets are harmonic.  CHAPTER 3. 3 7KLV WDEOH VKRZV WKH GHSHQGHQFH RI WKH WRWDO JHQHUDOL]HG VROLG DQJOH I(0), DQG WKH GHJHQHUDF\ RI WKH K\SHUVSKHULFDO KDUPRQLFV w , RQ WKH DUH FRQVWDQWV ZKLFK DSSHDU LQ WKH VXP UXOH IRU GLPHQVLRQ d .

15). &KDSWHU  +<3(563+(5,&$/ +$5021,&6 +DUPRQLF SRO\QRPLDOV In the previous chapter, we saw that when 3-dimensional momentum space is mapped onto the surface of a 4-dimensional hypersphere by Fock’s projection, the Fourier-transformed hydrogenlike atomic orbitals are represented by 4-dimensional hyperspherical harmonics, multiplied by a universal factor, 0 ( S )  which is independent of the quantum num bers. We mentioned also that our aim will be to use the momentumspace hydrogenlike functions as basis sets for constructing atomic and molecular orbitals, as well as crystal orbitals.

For more details concerning these relationships, the interested reader is referred t o the Russian literature and t o the papers of Aquilanti and his coworkers. 1 Decomposition of monomials into harmonic polynomials The indices i j k are assumed to be all unequal. The terms in brackets are harmonic.  CHAPTER 3. 3 7KLV WDEOH VKRZV WKH GHSHQGHQFH RI WKH WRWDO JHQHUDOL]HG VROLG DQJOH I(0), DQG WKH GHJHQHUDF\ RI WKH K\SHUVSKHULFDO KDUPRQLFV w , RQ WKH DUH FRQVWDQWV ZKLFK DSSHDU LQ WKH VXP UXOH IRU GLPHQVLRQ d .

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