Spin-coupled calculations based on projected spin eigenfunctions

1998 ◽  
Vol 99 (1) ◽  
pp. 64-67 ◽  
Author(s):  
Britt Friis-Jensen ◽  
David L. Cooper ◽  
Sten Rettrup
Keyword(s):  
Spin Eigenfunctions

The valence-bond theory of molecular structure II. Reformulation of the theory

1954 ◽  
Vol 223 (1154) ◽  
pp. 306-323 ◽  
Keyword(s):  
Valence Bond ◽  
Closed Shell ◽  
Numerical Illustration ◽  
Atomic Orbitals ◽  
Spin Eigenfunctions ◽  
Spin Pairing ◽  
Vb Structures ◽  
Bond Theory ◽  
Strong Repulsion ◽  
A Charge

The construction of spin eigenfunctions and the evaluation of matrix elements between ,them are discussed generally in preparation for a development of the valence bond (VB) theory along the lines indicated in I. The customary approximation of considering explicitly only the electrons outside a ‘closed shell’ is shown to be defensible. The reformulation of the VB theory is now straightforward, but its final description of bonding is quite new. Atomic orbitals (AO’s) are replaced, whenever they appear, by orthogonalized atomic orbitals (AO’s); but when the assumptions of the conventional theory are rigorously validated in this way the ‘covalent’ structures (now ‘VB’ structures) are found, quite generally, to indicate only strong repulsion between the ‘bonded’ atoms, and formal descriptions of bonding and of bond orders, in terms of ‘spin-pairing’, become nonsensical. Bonding can be described only by admitting into the wave functions polar VB structures; a bond between two atoms demands the appearance (with considerable weight) of pairs of structures differing by a ‘charge hop’ between the atoms concerned. The conventional VB structures are found to be equivalent to certain groupings of VB structures (non-polar and polar) and do, indeed, predict bonds between spin-paired atoms and repulsion between the atoms of different pairs. It is then possible to make full use of chemical intuition, using a plausible combination of conventional structures as a starting approximation in the more rigorous theory. A numerical illustration is provided by a discussion of the Kekulé structures of benzene. Some important characteristics of energy calculations in the VB theory are pointed out. Quantities of intra - and inter -atomic origin are well separated, and the method is apparently well suited to development along either ab initio or empirical lines.

SPINS: A collection of algorithms for symbolic generation and transformation of many-electron spin eigenfunctions

1995 ◽  
Vol 90 (1) ◽  
pp. 51-73 ◽  
Author(s):  
Peter B. Karadakov ◽  
Joseph Gerratt ◽  
David L. Cooper ◽  
Mario Raimondi
Keyword(s):  
Electron Spin ◽  
Spin Eigenfunctions

Die Pauli-Gleichung mit Differentialoperatoren für den Spin/ The Pauli Equation with Differential Operators for the Spin

1978 ◽  
Vol 33 (10) ◽  
pp. 1133-1150
Author(s):  
Eberhard Kern
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Differential Operators ◽  
Spin Operator ◽  
Rigid Bodies ◽  
The Body ◽  
Wave Functions ◽  
Pauli Equation ◽  
Spin Eigenfunctions ◽  
Cartesian Coordinate

The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result.The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory.The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.

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