Identifying a pair of interacting chromophores by using SVD transformed CIS wave functions
Graphical abstract
The manifestation of two similar chromphores is considered for the case of SVD transformed CIS wave functions.
Introduction
In a recent Letter [1] it was described that any individual CIS solution can be dramatically simplified by using the singular value decomposition (SVD) method: after transformation it contains no more than n = min(nocc, nvirt) excitations. For that reason one should determine the appropriate nocc × nocc and nvirt × nvirt orthogonal (real unitary) matrices U and V, respectively, which transform the nocc × nvirt CIS coefficient matrix Cσ to a generalized diagonal form asHere we have assumed n = nocc < nvirt; if the opposite is true, then matrix Λ has a form transposed to that is shown above, and one has n = nvirt. Superscript σ (=α or β) indicates that we consider excitations of orbitals with spin σ. Note that the matrices U and V providing this generalized diagonalization can always be determined by standard linear algebraic techniques (e.g., [2]), as used in our program [3].
In terms of the original occupied and virtual canonic HF orbitals , , the part of the CIS wave function which corresponds to excitations with spin σ can be written by using second quantization as(We indicate spin σ at the summations but not at the creation and annihilation operators and , respectively, in order to avoid very cumbersome notations.) By using the SVD method it is transformed to the form [1]where the creation and annihilation operators correspond to some new orthonormalized orbitalsobtained by performing separate unitary transformations within the occupied and virtual subspaces, respectively. Most recently Surján has showed [4] that these orbitals are nothing else than the natural orbitals of the multideterminant CIS wave function. This fact has a great importance for the considerations discussed below.
In this formalism the quantities λk play a central role: the relative weights of the individual determinants in the CIS wave function is given by . (The factor 2 appears because the CIS wave function contains excited determinants for both spins, and their sum is usually assumed to be normalized to one, i.e., each is normalized to .) One may expect (or at least hope) that only one or a very few values differ significantly from zero, thus one or a few excitations dominate in every CIS wave function.
The first exploratory calculations indeed confirmed that expectation and indicated that the CIS solutions often are dominated by a single excitation exhibiting a relative weight somewhere between 90% and 99%. However, a first preliminary study [5] into a system which is expected to exhibit CD (circular dichroism) activity in its electronic spectrum, indicated that at least two excitations contribute significantly to some of the CIS solutions. This is due to the presence of at least two chromophores in the same molecule – CD spectra are often connected to a pair of interacting chromophores. Special effects may take place if the two chromophores are identical (very similar). Accordingly, the present letter is devoted to the problem of analyzing how the presence of a pair of weakly interacting similar chromophores in a molecule manifests in the framework of the present formalism, and how one can extract from the SVD transformed results some further information concerning the individual chromophores.
Although both approaches involve unitary transformations of the occupied and virtual orbitals, our scheme has little in common with the direct use of localized orbitals in discussing excitation processes [6], [7], [8]; however, at the end of our analysis we shall arrive to orbitals which can be expected not too far from those from which these authors are starting. It may also be mentioned that, by using a different formalism, the authors of Ref. [9] have also, in fact, derived the orbitals (4) but stopped just a step short from realizing that they correspond to transitions as described in (3).
Section snippets
Idealized case of non-interacting chromophores
Let us assume that the system contains two independent identical chromophores the interaction between which is negligible. Let each chromophore is characterized by its pair of occupied and virtual orbital and , respectively. (Subscript i = 1 or 2 distinguishes the chromophores.) As noted above, the CIS wave function (3) obtained by using the SVD transformation is built up of the natural orbitals [4]. The natural orbitals of the CIS wave functions should correspond to the symmetry of the
Realistic case of weakly interacting chromophores
In the practice one does not have such idealized cases as discussed in the previous section, in which the CIS wave function consists of only two excitations of equal weights, as described by Eq. (6). One may, however, find cases in which the CIS wave function is dominated by a pair of related excitations:The larger the λi values are and the smaller is their difference, the idealized case is better approached. The deviation of the sum from the value
Identifying a pair of interacting chromophores: a sample calculation
According to the discussion in the previous section, one may identify a pair of interacting similar chromophores – in a chiral environment they may give rise to CD activity – on the basis of CIS calculations in the following manner.
One has to look for a pair of CIS solutions with a close energy, which satisfy the conditions:
- (a)
after the SVD transformation have (at least) two λ values significantly differing from zero. The larger are the λ values and the smaller is their difference, the idealized
Acknowledgements
The author is grateful to Prof. P.R. Surján for his useful comments on the manuscript and to Profs. J.G. Ángyan and G. Lendvay for calling his attention to Refs. [6], [7], [8], [9].
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