Total spin angular momentum operator

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

L the total orbital angular momentum operator the total spin angular momentum operator... 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

In a many-electron system, the total spin angular momentum operator is simply the vector sum of the spin vectors of each of the electrons... 

After separating the quadratic term into expressions containing individually the total orbital angular momentum operator L and the total spin angular momentum operator S, it can be shown that [55]... 

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... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

Here, as elsewhere, the subscripts i and a stand for electrons and nuclei respectively. The factor (S Si)/S(S + 1) is used to project the contribution from each open shell electron i onto the total spin angular momentum S. We remind ourselves that the effective Hamiltonian is constructed to operate within an electronic state with a given multiplicity (2S +1). 

The wavefunction must be an eigenfunction of the operators corresponding to the square of the total spin angular momentum (S2) and to its z component S2.f... 

Analogous to the orbital angular-momentum operators L, L L L, we have the spin angular-momentum operators S, S, Sy, S which are postulated to be linear and Hermitian. is the operator for the square of the magnitude of the total spin angular momentum of a particle. is the operator for the z component of the particle s spin angular momentum. We have... 

UHF calculations consequently give different spatial orbitals for a- and j6-spins. This causes a problem in that UHF wavefunctions are often not eigenfunctions for the square of the total spin operator, S. The total spin operator S is the sum of the spin angular momentum operators of the constituent electrons, s, and is written as... 

Up = ordinary Hall coefficient R, = spontaneous Hall coefficient R, = extraordinary Hall coefficient 5 = spin quantum number S - total spin angular momentum 5j = spin operators T = absolute temperature T = tesla... 

The iV-electron operator fh RGo) is called the orbital magnetic dipole operator, kiRoo) is the orbital angular momentum operator of electron i with respect to the gauge origin Rqo and Sj is the spin angular momentum operator of electron i. The total orbital and spin angular moment operators of all N electrons are... 

For the square of the total spin angular momentum the operator is... 

We can operate quite analogously with spin angular momentums. As i = 1/2 and N is the number of electrons, the total spin angular momentums S can accept the values... 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

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