Alpha spins

A simple measure of the election density distribution over the participating atoms is the Mulliken population [60]. For linear Li—H—Li the alpha spin is... 

Spin orbitals arc grouped in pairs for an KHF ealetilation, Haeti mem her of ih e pair dilTcrs in its spin function (one alpha and one beta), hilt both must share the same space function. For X electrons, X/2 different in olecu lar orbitals (space function s larc doubly occupied, with one alpha (spin up) and one beta (spin down) electron forming a pair. 

HyperChem tjuantum tn ech an ics calcu lation s tn ust start with the number of electrons (N) and how many of them have alpha spins (th e remain in g electron s have beta spin s ). HyperCh em obtain s th is in form ation from the charge an d spin m u Itiplicity th at you specify in th e Sem i-em pirical Op lion s dialog box or. Ab Initio Option s dialog box. is th en computed by coun ting the electron s (valence electrons in sem i-em pirical methods and all electrons in a/ irti/io m ethod) associated with each (assumed neutral) atom and... 

Con versely, an imre.vtrtctcrf Hartree-Fock description implies that there are two different sets of spatial molecular orbitals those molecular orbitals, occupied by electrons of spin up (alpha spin ) and those molecular orbitals, occupied by electrons of spin down (beta spin) as shown next. 

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin ... 

This is more natural, since our intuition is usually based on having a region of space which describes the location (more or less) of two electrons, one of alpha spin and one of beta spin. Some of quantum chemistry is formulated entirely in terms of spin orbitals, for various reasons. For our purposes, we will work entirely in the spatial orbital basis. This will cause things to get somewhat murky soon, but in the long run it will be simpler. 

As the MSADs have the largest coefficients in the ground-state wave function, one may expect that for alternant free radicals or polyradicals, the dominant positive spin density will be largest on the atoms that bear an alpha spin in the MSAD. An example that was analyzed in Chapter 7 is the allyl radical, where the MSAD predicts positive spin densities at positions 1, 3 and a negative density at position 2. Similarly, the MSAD of benzyl radical predicts positive spin densities on the benzylic carbon and on the ortho and para positions. 

Cilice 0 < c < 1 the first term shows tliat UHF orbitals reduce the ionic coutribution relative to the covalent structures, compared to the RHF case, eq. (4.18). This is the same effect as for the Cl procedure (eq. (4.19)), i.e. the first term shows that the UHF wave function partly includes electron correlation The fii t term can he written as a linear combination of the 0 and 1 determinants, and describes a pure singlet. state. The last part of the UHF determinant, however, has terms identical to two of those in the triplet combination (4.22). If we had chosen the alternative set of UHF orbitals A ith the alpha spin being primarily on centre B in eq. (4.20), we would have obtained the other two terms in i.e. the last term in (4.24) breaks the symmetry. The UHF determinant is therefore not a pure spin state, it contains both singlet and triplet spin ates. This feature is known as spin contamination. Foi c — 1 the UHF wave function i... 

Although Handy was the first to use alpha and beta strings, we will employ the subsequent notation of Olsen et al.46 An alpha string is defined as an ordered product of creation operators for spin orbitals with alpha spin. If Ia contains a list, j,... A of the Na occupied spin orbitals with alpha spin in determinant 7), then the alpha string a(Ia) is ajQaja... aj.a. A beta string is defined similarly. Thus a Slater determinant /) in terms of alpha and beta strings is... 

Direct Cl methods often require an index vector which points to a list of all allowed excitations from a given iV-electron basis function. Using alpha and beta strings, the index vector need not be the length of the Cl vector—its size is dictated by the number of alpha or beta strings, which (for a full Cl) is approximately the square root of the number of determinants. This results from the fact that in determinant-based Cl, electrons in alpha spin-orbitals can be excited only to other alpha spin-orbitals, and electrons in beta spin-orbitals can be excited only to other beta spin-orbitals (because of the restriction to a single value of Ms). 

The determinants in the Cl expansion therefore have Na electrons of alpha spin and electrons of beta spin. With the alpha spin orbitals preceding the beta spin orbitals, each Slater (teterminant may be written as... 

The MOC algorithm performs poorly when the number of electrons of alpha spin is small (one or two). With a single alpha electron, the only string for which a apa I ) is nonzero is la = a a vac). The operation count for (11.8.42) then becomes comparable with the operation count for (11.8.43). Moreover, each element is only used once in (11.8.43) and the... 

For simplicity, we here transform only the orbitals that have alpha spin. The orbitals with beta spin may be included in the vector (11.9.3) with no complications except of notatirm. In each step, we generate a new set of determinants and determine a new set of Cl coefficients such that the identity... 

HftTURAL BOND ORBITAL AKALYSIS, alpha spin orbitals ... 

The general principles of NBO donor-acceptor interactions (Chapter 5) suggest the strong delocalizations to be expected for the triplet-state Lewis structures (11.22). In the alpha spin set, the allylic-like C=C I pattern immediately suggests strong nc n cc delocalization, and the anionic -O oxo pattern suggests enhanced... 

Next, we note that different determinants may have the same orbital occupation numbers but different spin-orbital occupation numbers since an orbital occupation equal to 1 may represent either the occupation of an alpha spin orbital or the occupation of a beta spin orbital by an unpaired electron. The set of all determinants with the same orbital occupation numbers but different spin-orbital occupation numbers is said to constitute an orbital configuration. In a sense. 

There is here one set of parameters for the alpha spin orbitals and one set for the beta spin orbitals. Indeed, these are the parameters used in UHF theory for carrying out an unrestricted optimization of the wave function, generating separate sets of alpha and heta spin oibitals. 

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