Genealogical coupling

We now consider the expansion of a given N-electron genealogical CSF with total spin S and projected spin M in determinants with spin projection M belonging to the same orbital configuration  

Each determinant may be written as a product of creation operators working on the closed shell of the spin-paired core electrons  

Here we enumerate the singly occupied orbitals by integers from 1 to Ai. The spin projection of each spin orbital is obtained from the corresponding element of p. To determine the coefficients of the expansion (2.6.1), it is convenient to rewrite the CSF as a tensor operator working on the closed shell of core electrons  

The operator (t) generates a normalized 7V-electron spin eigenfunction as specified by the genealogical vector t. Since CSFs may be written as linear combinations of Slater determinants, the tensor r rator in (2.6.3) must be a linear combination of operator strings each containing N creation operators. The tensor operator therefore commutes with creation operators for even N and anticommutes for odd N. 

In the genealogical scheme, we envisage each //-electron state [t) as arising by a coupling of the one-body alpha and beta creation operators with a spin-adapted (N — 1 )-electron state  

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Singlet excitation operators may be generated by the genealogical coupling of doublets of creation operators a p, aj, and annihilation operators —Opp, Opa] as described in Section 2.6.7. In the spin-orbital basis, the excitation operators Xp, contain the inactive annihilation doublets —a,, fl,a and the virtual creation doublets a , a p). To obtain excitation operators of singlet symmetry, we must, for the active orbitals, include the pairs a p and —Ovp, a . However, for the high-spin... 

We give in this section an introduction to the construction of CSFs and more generally to the construction of spin tensor operators. We shall employ the genealogical coupling scheme, where the final CSF for N electrons is arrived at in a sequence of N steps [2]. At each step, a new electron is introduced and coupled to those already present. We thus arrive at the final CSF through a sequence of N —I intermediate CSFs, each of which represents a spin eigenfunction. 

The genealogical coupling coefficients in (2.6.4) are determined from the requirement... 

Note that the genealogical coupling coefficients are independent of the number of electrons. 

Once our N-electron CSF has been reduced to a linear combination of one or two (N — 1)-electron spin eigenfunctions each multiplied by a creation operator, we may go one step further and expand each (N — l)-electron state in two (N — 2)-electron spin eigenfunctions as dictated by the penultimate element t -i in the genealogical coupling vector t. After N — I such steps, we arrive at an expansion in terms of determinants with projected spin M. In this way, we are led directly to an expansion of CSFs in Slater determinants where the coefficients are products of the genealogical coupling coefficients in (2.6.5) and (2.6.6). 

Figure 8. (B) In two-dimensional model genealogy is represented by two-dimensional spin lattice. Individual spins exist in two states (J and J.) corresponding to two digits in binary sequences. Every row of lattice consists of v digits and corresponds to polynucleotide sequence. In-row interaction, described by spin-spin coupling constant is property of individual sequence and contributes to rate constant of replication. Vertical" coupling constant J, on the other hand, is measure of mutation frequency.

Negishi, E.-i. A genealogy of Pd-catalyzed cross-coupling. J. Organomet. Chem. 2002, 653, 34-40. 

Miura, M., Nomura, M. Direct arylation via cleavage of activated and unactivated C-H bonds. Top. Curr. Chem. 2002, 219, 211-241. Negishi, E.-i. A genealogy of Pd-catalyzed cross-coupling. J. Organomet. Chem. 2002, 653, 34-40. 

As an internal A-electron basis X > we use genealogical spin eigenfunctions. The one-particle coupling coefficients yfj = are evaluated using well known group theoretical methods and stored on disc. We first consider the coefficients needed to calculate the a ijp, kip) and fiiijp, klq)... 

Calculations.—Molecular orbital theory calculations of complexes of borane (BH3) with such molecules as dimethyl ether, nitriles and isonitriles, and ammonia have been made. Ab initio calculations on systems of the types XNH3+, XCH3, and XBHs have been used to develop a theoretical approach to substituent effects.A genealogical electronic coupling procedure has been applied in calculations of the excited states of borane. ... 

In general, both alpha and beta spin orbitals contribute to this coupling. We therefore have a sum over two (N — l)-electron states in (2.6.4), one with spin projection M — (coupled to an alpha electron) and one with projection M + (coupled to a beta electron). The total spin is Tjv-i = S — tN for each (N — l)-electron state, where tf is the last element in the genealogical vector. The coupling coefficients in (2.6.4) depend on the total and projected spins of the coupled state 5 and M and also on the spins of the creation operator a and 1 , both of which may take... 

Show that the number of distinct genealogical spin couplings with Nopen unpaired electrons and spin S is given as... 

In this exercise, we use the genealogical scheme to couple the tensor operators a pp] and —Oqp, aqa) to Obtain spin-adapted one-electron operators. 

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