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Spin-functions

Construction of Spin Functions.—It can be seen from expressions (26) and (27) that, besides the various integrals required, the basic group theoretical quantities which one needs are the matrices US(P). The form of these in turn is determined by the way the set of spin functions of equation (9) are constructed. We note that the form of the wavefunction (7) is unchanged by any simultaneous unitary transformation of the functions 0sk and There is [Pg.65]

The simplest method of constructing the functions M. t is by coupling together successively the spins according to the usual rules for coupling angular momenta in quantum mechanics. The index k on the spin functions in this basis may then be thought of as a set of partial resultant spins, [Pg.65]

For a further description of this basis and how the US(P) matrices in it are constructed, the reader is referred to the article by Kotani et al.s [Pg.65]

Another basis of some importance is one in which two standard functions of Ni and Nz electrons, respectively, are coupled together  [Pg.66]

The standard basis and equation (28) are connected by an orthogonal transformation of the form [Pg.66]

The spin orbitals introduced in Chapter 1 are functions of three continuous spatial coordinates r and one discrete spin coordinate nts- The spin coordinate takes on only two values, rqtresenting the two allowed values of the projected spin angular momentum of the electron = — and wi =. The spin space is accordingly spanned by two functions, which are taken to be the eigenfunctions a ms) and (ms) of the projected spin angular-momentum operator [Pg.34]

These spin functions - which we shall generically denote by a, r, ju and v - are eigenfunctions of the total-spin angular-momentum operator as well with quantum number s = 5 [Pg.34]

Completeness of the spin basis leads to the following resolution of the identity [Pg.34]

We shall occasionally find it convenient to use for the discrete spin functions the same notation as for continuous spatial functions, interpreting integration in spin space as summation over the two discrete values of ms . [Pg.35]

we may write the orthonormality conditions of the spin functions in the form [Pg.35]


The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin-orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms. [Pg.392]

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... [Pg.551]

As is well known, when the electronic spin-orbit interaction is small, the total electronic wave function v / (r, s R) can be written as the product of a spatial wave function R) and a spin function t / (s). For this, we can use either... [Pg.560]

Species of Spin Functions for Some Important Double Groups... [Pg.563]

In summary, for a homonuclear diatomic molecule there are generally (2/ + 1) (7+1) symmetric and (27+1)7 antisymmetric nuclear spin functions. For example, from Eqs. (50) and (51), the statistical weights of the symmetric and antisymmetric nuclear spin functions of Li2 will be and respectively. This is also true when one considers Li2 Li and Li2 Li. For the former, the statistical weights of the symmetric and antisymmetiic nuclear spin functions are and, respectively for the latter, they are and in the same order. [Pg.571]

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... [Pg.574]

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. [Pg.575]

However, drastic consequences may arise if the nuclear spin is 0 or In these cases, some rovibronic states cannot be observed since they are symmetry forbidden. For example, in the case of C 02, the nuclei are spinless and the nuclear spin function is symmetric under permutation of the oxygen nuclei. Since the ground electronic state is only even values of J exist for the ground vibrational level (vj, V3) = (OO O), where (vi,V2,V3) are the... [Pg.580]

For molecules with an even number of electrons, the spin function has only single-valued representations just as the spatial wave function. For these molecules, any degenerate spin-orbit state is unstable in the symmetric conformation since there is always a nontotally symmetric normal coordinate along which the potential energy depends linearly. For example, for an - state of a C3 molecule, the spin function has species da and E that upon... [Pg.603]

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. [Pg.605]

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs of the 3D rotational group SO(3), where a is the rotational vector, written in the operator form as [36]... [Pg.619]

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. [Pg.37]

X spin orbital (product of spatial orbital and a spin function)... [Pg.15]

A determinant is the most convenient way to write down the permitted functional forms of a polv electronic wavefunction that satisfies the antisymmetry principle. In general, if we have electrons in spin orbitals Xi,X2, , Xn (where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is ... [Pg.59]

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... [Pg.145]

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... [Pg.280]

The spin functions a and P which accompany each orbital in lsalsP2sa2sP have been eliminated by carrying out the spin integrations as discussed above. Because H contains no spin operators, this is straightforward and amounts to keeping integrals <( i I f I ( j > only if ( )i and ( )j are of the same spin and integrals... [Pg.285]

To avoid having the wave function zero everywhere (an unacceptable solution), the spin orbitals must be fundamentally different from one another. Eor example, they cannot be related by a constant factor. You can write each spin orbital as a product of a space function which depends only on the x, y, and z coordinates of the electron—and a spin function. The space function isusually called the molecular orbital. While an infinite number of space functions are possible, only two spin functions are possible alpha and beta. [Pg.36]

To use HyperChem for calculations, you specify the total molecular charge and spin multiplicity (see Charge, Spin, and Excited State on page 119). The calculation selects the appropriate many-electron wave function with the correct number of alpha or beta electrons. You don t need to specify the spin function of each orbital. [Pg.36]


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A. spin function

Anti-symmetric nuclear spin function

Antisymmetric spin functions

Cartesian triplet spin functions

Configurational function spin-free

Configurational spin functions

Density Functional Theory spin potential

Density function theory spin-dependent properties

Density functional theory spin-orbit effects

Density functional theory-electron spin resonance calculations

Deuteron nuclear spin functions

Eigenfunction spin functions

Electron-spin spectral density functions

Function coupled spin

Gaussian functions, spin-orbit operators

Generalized gradients spin functionals

Generation of Many Electron Spin Functions

Ground-state wave function electronic Hamiltonian, spin-orbit

Irreducible representations nuclear spin function

Kotani spin function

Local spin density functional

Local spin-density approximations hybrid exchange functionals

Local spin-density functional theory

Local spin-density functional theory applications

Natural spin-orbital functional

Natural spin-orbital functional characterized

Nuclear Spin Basis Functions

Nuclear Spins and Wave Function Symmetry

Nuclear spin additional functions

Nuclear spin function

Nuclear-spin partition function

P, spin function

Permutational symmetry nuclear spin function

Properties of Spin Functions

Pure- and Mixed-spin Wave Functions

Reduced density-functions spin factors

Response functions spin-orbit

Rumer basis, spin functions

Rumer spin functions

Scalar-relativistic/spin-free function

Slater-type functions, spin orbital products

Spectral function spin-boson model

Spin Degeneracy and Wave-Functions for Increased-Valence Structures

Spin autocorrelation functions

Spin correlation function

Spin density functional methods

Spin function, permutational symmetry

Spin functionals, gradients

Spin functions a and

Spin functions general

Spin functions singlet and triplet

Spin orbitals representation, functionals

Spin partitioning functional

Spin restricted wave function

Spin spinor function

Spin wave function

Spin-coupled wave function

Spin-coupled wave function determination

Spin-density functional theory

Spin-density functional theory nonrelativistic

Spin-density functionals

Spin-free density function

Spin-lattice relaxation oxidized functional groups

Spin-orbit operators functions

Spin-orbit perturbed wave functions

Spin-orbitals orthonormalized functions

Spin-polarized density functional theory

Spin-polarized density functional theory chemical reactivity

Spin-polarized density functional theory energy function

Spin-potential in density functional theory framework

Spin-restricted open-shell Hartree-Fock ROHF) reference functions

Spin-velocity density function

Summary of Kohn-Sham Spin-Density Functional Theory

Symmetric properties nuclear spin function

Symmetric spin functions

Symmetry of spin wave functions

Symmetry spin functions

The spin functions

Two-electron spin functions

Wave function mixed-spin state

Wave function nuclear spin

Yamanouchi-Kotani spin functions

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