182 Semiempirical MO-CIS approaches to energy transfer
4.2 Semiempirical MO-CIS approaches to energy
transitions will only involve MO localized on either theDorAunit. Therefore, it is easy to select the two states responsible for the for the RET process, nominally the D excited state (when A is in the ground state) and the A excited state (when Dis in the ground state configuration); respectively written as |D∗A0〉 and |D0A∗〉. In the CI approach these are expressed as as linear combination of single excited configurations (compare with equation C.7):
|D∗A0〉∞ = X
i,r
C∞
Di,rΨi→r (4.15)
|D0A∗〉∞ = X
i,r
C∞
Ai,rΨi→r (4.16)
where C∞
Di,r (C∞
Ai,r) are the CI coefficients relevant to the |D∗A0〉 (|D0A∗〉) state described at infinite distance, and Ψi→ris the configuration obtained promoting an electron from the i-th occupied MO towards the r-th virtual MO. Here and in the following we will adopt the frozen orbital approach, and whenever MO are introduced they refer to infinite distance.
The interaction V is then calculated (according to its perturbative defini-tion), on the basis of these frozen MOs referring to infinite distance, but actually using the finite distance Hamiltonian Hd. According to this scheme, the relevant matrix element for RET interaction then reads:
V=∞〈D∗A0| ˆHd|D0A∗〉∞=X
ir, js
C∞
Di,rC∞
Aj,s〈Ψi→r| ˆFd|Ψj→s〉 =X
ir, js
Ci,r∞C∞j,sHir, jsd (4.17) where ˆFd is the Fock matrix operator for the pair of molecules at distance d.
The approach is expected to apply when d is small enough for the two molecules to feel each other, and large enough to neglect any intermolecular charge-transfer. According to eq. 4.17 the calculation of the RET matrix ele-ments is possible provided that we have expressions for the coefficient of the CI expansion of basis (infinite distance) states (Ci,r∞ and related in eq. 4.15 and 4.16) and the elements of the CI matrix, Hd calculated at finite distance on the basis of the frozen orbitals.
The calculation of Hir, jsd
To find explicit expression for the CI matrix elements on the frozen orbital basis we work in second quantization and label occupied and virtual orbitals of the
Dmolecules with the i and the r indexes, respectively. Symbols j and s refer to occupied and virtual orbitals of theAmolecule. The matrix elements relevant to RET, Hir, jsd in eq. 4.17, have i 6= j and r 6= s. The general molecular Hamiltonian,
184 Semiempirical MO-CIS approaches to energy transfer
on the basis of the MO, can be written as:
H =X
k,l,σ
hkla†k,σal,σ+1 2
X
klmn,σ,µ
(kn|lm)ak,σ† a†l,µam,µan,µ (4.18)
where Greek letters refer to AO, hkl are the matrix elements of the one-electron Hamiltonian and (kn|lm) are the bielectronic repulsion terms. Moreover:
Φi→r = 1
p2[a†1,αa1,β† . . . a†i,α. . . a†r,β
−a†1,αa1,β† . . . a†i,β. . . a†r,α]|0〉 (4.19) Φj→s = 1
p2[a†1,αa1,β† . . . a†j,α. . . a†s,β
−a†1,αa1,β† . . . a†j,β. . . a†s,α]|0〉 (4.20) For the CIS matrix elements of interest for RET (i 6= j and r 6= s) only bielectronic terms are relevant and the resulting expression, derived from equation 4.17, actually coincides with the standard expression built with Slater’s rule (p. 236 in [153]), and then:
Hdir, js=2(ri|js) − (rs|ji) (4.21)
The equivalence of eq. 4.21 with the standard expression was not granted from the beginning. Indeed the standard expression for the CIS Hamiltonian is writ-ten on the basis of determinants that are built with MOs that diagonalize the Fock matrix. The MO that appear in eq 4.17 do not diagonalize the Fock matrix at distance d, while they diagonalize the Fock matrix at infinite distance. The formal equivalence is related to the fact that the only matrix elements of Hd in the RET refer to the non diagonal elements that mix the configurations coming from theDexcitations with the one fromA. The same equivalence does not ap-ply to diagonal elements of the CI matrix. The four-center integrals that enter the equation 4.21 are the same that enter a regular CIS calculation, and then can be calculated in the INDO/S approach, according to the approximations and the expressions reported in the appendix sect. C.3.2, and imposing frozen orbitals.
Therefore, to calculate V within this INDO/S MO–CIS formulation, the cal-culation has to be run for the pair DandA first at very large distance, to cal-culate relevant MO, select relevant excitations, and obtain single excited con-figuration coefficients C∞
Di,r ;Aj,sof eq. 4.15 and the frozen MOs. Thereafter, a calculation is run at finite d distance, without diagonalizing any Fock or CIS ma-trix, but simply evaluating Hir, jsd according to eq. 4.21. This information enters eq. 4.17 to get the required RET matrix element. Thus, no diagonalizations at
finite distance are required: only bielectronic integrals (ri|js) and (rs|ji) have to be calculated at finite d for the calculation of selected matrix elements of the CI matrix, in number of 2NC, in NC is the number of configurations that are basis for the CI. The calculation of four center integrals, even if approximated in the INDO/S Hamiltonian, is actually computationally demanding when the complete CIS is needed, as required for the calculation of excitation at infinite distance. Actually the infinite distance calculation can be optimized by perform-ing the calculation on the two molecules separately and diagonalizperform-ing the Fock and CIS matrix for theDandAseparately, obtaining the relevant MO and exci-tations. Then the AO basis for the supermolecule can be constructed by simply queuingAAOs afterDAOs. The MOs and the CIS matrix for the supermolecule can be easily rewritten on the composite basis, filling with zeroes. Then the calculation of Hir, jsd can be performed normally. In these calculation at finite distance d, only the specific CI matrix elements entering the expression for V (eq. 4.17) have to be calculated, representing just a minimal fraction of the full CI matrix. For the calculation of V(d) we have modified an available INDO/S code [130], (see appendix C.3).
4.2.2 From transition densities to point atomic charge densi-ties in the INDO/S
For the sake of comparison we elaborate on the transition density approach described in sect. 4.1.1 and discuss how it applies to INDO/S Hamiltonian. In particular, as in other semiempirical approaches, in INDO/S the basis wavefunc-tions are not fully defined (i.e. they do not have an explicit space coordinate dependence), so that the transition densities as defined in eq. 4.6 and 4.7 can-not be calculated. Transition densities in fact collapses in INDO-like approaches into a distribution of transition charges located at atomic sites. Then we define point transition charges, located at the atomic position. For a single configura-tion we will refer to the i → r excited configuraconfigura-tion forDand j → s forAunit in the way that the configuration transition densities are:
qi→rd=X
δ∈d
ciδcrδ (4.22)
qj→sa=X
α∈a
cjαcsα (4.23)
where the first equation stand for theD molecule and δ refers to all the AO centered on the atom d, while the second equation stands for the moleculeA(α is the AO for a atom). The coefficients in equation 4.22 are the MO coefficients.
186 Extended dipole methods in essential state models
Once the transition is defined as a linear combination of single CI (see appendix eq. C.8, and also 4.15), the point transition charge density results:
˜qdD
∗
A
0=X
(i,r)
CDi,rqi→rd (4.24)
˜qaD
0
A
∗=X
(j,s)
CAj,sqj→sa (4.25)
where again the first line refer to the transition charge at the atom d for an excitation localized on the donor and the second line refer to the localized ac-ceptor excitation. The above expressions for the charge densities refer to the non-interactingD−Apair. The calculation, that involve extensive CIS, can be conveniently performed separately onDand onA.
In this approximation the RET interaction energy in 4.8 reads:
V=X
d
X
a
˜qdD
∗
A
0˜qaD
0
A
∗
1
|rd− ra| (4.26)