Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S² eigenfunctions

Abstract

We present an implementation of the spin-dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry-adapted functions.Using S 2 eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin-independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one-and two-electron charge densities. K E Y W O R D S configuration interaction, configuration state function, quantum chemistry, spin density, unitary group approach 1 | I N TR ODU C TI ONThe unitary group approach (UGA) in quantum chemistry was pioneered by Paldus in 1974, [1] with practical algorithms including the most famous graphical approach by Shavitt, [2,3] having been developed in the following years. It offered a simple and straightforward way of evaluating the spinless generator matrix elements (ME) using implicit Gelfand-Tsetlin basis [4,5] and was suitable for calculating many-electron wavefunction expectation values for spin-independent Hamiltonians. [6] The growing demand for relativisitc corrections and other spin dependent properties in wavefunction methods inspired the subsequent development of spin-dependent generator ME in 80-90s by the several groups of authors [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the latest work in this field is actually quite recent. [24] In particular, a theory for spin dependent operators formulated entirely within the unitary group formalism has been developed by Gould, Chandler, Paldus, and Battle. [7][8][9][10][11][12][13][14][15] As we shall discuss, the final formulae are easy to implement and are based on spin-independent U(n) and well-known U(2) generator ME in terms of the Gelfand-Tsetlin implicit basis. As a special, zero-order case, this theory provides a rather simple way for calculating spin density. [13,15,25] An obvious approach to spin density calculation with configuration interaction (CI) wavefunction would be to use the precomputed spindependent generator ME. The general single-particle reduced density operator (in a 2 3 2 matrix form) is given by [13] : qðr; r 0 Þ lm 5 X n i;j51

Charge migration engineered by localisation: electron-nuclear dynamics in polyenes and glycine

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Using ‘designer’ coherences to control electron transfer in a model bis(hydrazine) radical cation: can we still distinguish between direct and superexchange mechanisms?

“…We shall now focus on the electron dynamics prompted upon initializing the cationic system in the state Ψ 2 , which involves the ϕ 2 , ϕ 3 and ϕ 4 adiabatic electronic states (see figure 4(b)). Looking at the coefficients of each ϕ i =2−4 state in table 1, it can be inferred that the period of the oscillation of the N spin densities is driven by the energy difference between E (ϕ 4 ) − E (ϕ 3 ) = 0.28 eV (τ 2 ∼ 15 fs), while the period of C atom electron density oscillation is driven by the (average) [19] onto N atomic sites or partial summations onto the coupler benzene ring. Specifically, (a) e-TS from Ψ 1 resembles the electronic structure of the ridge E-F in the direct mechanism (figure 2(a)), (b) e-TS from Ψ 2 resembles the quinoidal intermediate D of the superexchange mechanism but it is also related to the direct mechanism since the bridge electron dynamics is passive and non-perturbing, and (c) e-TS from Ψ 3 resembles the anti-quinoidal intermediate C of the superexchange mechanism (the ring dynamics has a significant spin density amplitude that oscillates between 0.20 and 0.03 following the 2 fs dynamics, as shown in figure 9, however, there is a partial cancellation every 2 fs starting from 1/8 τ 3 , which blurs the e-TS).…”

The quantum-Ehrenfest method with the inclusion of an IR pulse: Application to electron dynamics of the allene radical cation

“…The spin density provides us with information about the location of the lone electron and has the advantage that it is not dependent on the details of the electronic structure method. 37 The analysis of the propagation of the nuclear wavefunction is done in the normal mode displacement basis (averaged over the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.…”