Spin‐Dependent operators in the spin‐free quantum chemistry

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AbstractsThe Fock generating function method is employed for the separation of spin variables into two-particle transition density matrices. Their spatial components introduced by McWeeny are expressed through Schrodinger's coordinate wave functions. Some nontrivial integral relations between these components and the charge and transition spin density matrices are obtained. The interrelation between the mentioned spatial components and the Matsen-Poshusta symmetrized density matrices of an arbitrary spatial function is found.

Section: (4)mentioning

confidence: 99%

Section: Eqsmentioning

confidence: 99%

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Fock coordinate function method for separation of spin variables in transition density matrices

Klimko¹,

Mestechkin²,

Whyman³

1980

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“…On account of Eq. (46) (and the spin dependence of the second-order density matrix [23]) we obtain only three different eigenvalues: f2(2a + b")) < 3 corresponding to the triplet natural geminals (q,(1)q1(2) -q,(1)q1(2))/V3, f2(2a -b'") < 1 corresponding to the singlet geminals (q,(l)q1(2) + q,(l)qi(2))/V3 and f2(2a -b'" -mb'") = 5" = ( n -u ) ( m + 1 -n -u ) / m corresponding to a single ex-treme natural singlet geminal P( 1 I 2 ) / 6 . The closer AQM (44) is to the averaged state, the nearer this occupation number f' is to (n -s) * …”

Section: On the Hund Rule In Roothaan's Theorymentioning

confidence: 99%

“…On account of Eq. (46) (and the spin dependence of the second-order density matrix [23]) we obtain only three different eigenvalues: f2(2a + b")) < 3 corresponding to the triplet natural geminals (q,(1)q1(2)q,(1)q1(2))/V3, f2(2ab'") < 1 corresponding to the singlet geminals (q,(l)q1(2) + q,(l)qi(2))/V3 and f2(2ab'" -mb'") = 5" = ( nu ) ( m + 1nu ) / m corresponding to a single ex-treme natural singlet geminal P( 1 I 2 ) / 6 . The closer AQM (44) is to the averaged state, the nearer this occupation number f' is to (ns) * ( m + 1 -ns ) / m , which is the maximal occupation number of the state with spin s. For s = 0 AQM degenerates into the extremal antisymmetric geminal power with a maximal possible second-order occupation number r , approaching n for a larger m [22].…”

Section: On the Hund Rule In Roothaan's Theorymentioning

confidence: 99%

Roothaan's open shell theory from the viewpoint of an orthogonal group

Klimko¹,

Mestechkin²

1990

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Explicit equations for the constants a and b of the energy as the average expectation value for the minimal domain of states giving the Roothaan energy functional are derived in terms of the total spin value and the "seniority" quantum number. The state with off-diagonal long-range order can appear in the framework of the Roothaan scheme for the quasidegenerate system as an alternative to the Hund rule. Admissible many-electron states are established for all configurations of the icosahedral symmetry group systems, and corresponding a and b coefficients are calculated.

Spin‐free quantum chemistry: What one can gain from fock's cyclic symmetry

Luzanov

2010

ABSTRACT:The spin-free wave function due to Fock (Zh Eksp Teor Fiz, 1940, 10, 961) is re-examined with a stress on the reduced density matrix (RDM) theory. The key notion of the Fock approach is the cyclic symmetry of wave functions. It is a specific algebraic identity involving transpositions of numbers taken from two different columns of the corresponding Young tableau. We show first how to construct symmetry adapted states by accounting for high-order cyclic symmetry conditions. For Young's projectors, it gives a new expression including nothing but antisymmetrizers. Next, transforming the Fock spin-free state by a duality operator (the star operator in exterior algebra), we arrive at the representation closely related to spin-flip models. In such spin-flip models, a coupling operator is the basic object for which we show that the cyclic symmetry is transformed into a tracelessness of the coupling operator. The main results are related to the spin-free theory of spin properties. In particular, the theorem previously stated (Luzanov and Whyman, Int J Quantum Chem, 1981, 20, 1179) is refined by an explicit general representation of spin density operators through spin-free (charge) RDMs. Some applications implicating high-order RDMs (collectivity numbers, the unpaired electron problem, cumulant spin RDMs, spin correlators, etc.) are also considered. For spin-free RDM components, a new projection procedure without constructing any symmetry adapted state is proposed. An unsolved problem of constructing orthogonal representation matrices within the Fock theory is raised.