**182** **Semiempirical MO-CIS approaches to energy transfer**

**4.2** **Semiempirical MO-CIS approaches to energy**

transitions will only involve MO localized on either the^{D}or^{A}unit. 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 ^{D}is in the ground state configuration); respectively written as |^{D}^{∗}^{A}^{0}〉
and |^{D}^{0}^{A}^{∗}〉. In the CI approach these are expressed as as linear combination of
single excited configurations (compare with equation C.7):

|^{D}^{∗}^{A}^{0}〉∞ = X

*i,r*

*C*^{∞}

D*i,r*Ψ* _{i→r}* (4.15)

|^{D}^{0}^{A}^{∗}〉∞ = X

*i,r*

*C*^{∞}

A*i,r*Ψ* _{i→r}* (4.16)

*where C*^{∞}

D*i,r* *(C*^{∞}

A*i,r**) are the CI coefficients relevant to the |D*^{∗}*A*^{0}*〉 (|D*^{0}*A*^{∗}〉) state
described at infinite distance, and Ψ* _{i→r}*is 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 H** ^{d}*. According to this scheme, the relevant
matrix element for RET interaction then reads:

V=_{∞}〈^{D}^{∗}^{A}^{0}| ˆ*H** ^{d}*|

^{D}

^{0}

^{A}

^{∗}〉

_{∞}=X

*ir, js*

*C*^{∞}

D*i,r**C*^{∞}

A*j,s*〈Ψ* _{i→r}*| ˆ

*F*

*|Ψ*

^{d}*〉 =X*

_{j→s}*ir, js*

*C*_{i,r}^{∞}*C*^{∞}_{j,s}*H*_{ir, js}* ^{d}*
(4.17)

*where ˆF*

^{d}*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 (C*_{i,r}^{∞} and related in eq. 4.15 and
**4.16) and the elements of the CI matrix, H*** ^{d}* calculated at finite distance on the
basis of the frozen orbitals.

**The calculation of H**_{ir, js}^{d}

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

D*molecules with the i and the r indexes, respectively. Symbols j and s refer to*
occupied and virtual orbitals of the^{A}molecule. The matrix elements relevant to
*RET, H*_{ir, js}^{d}*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*

*h*_{kl}*a*^{†}_{k,σ}*a** _{l,σ}*+1
2

X

*klmn,σ,µ*

(*kn|lm)a*_{k,σ}^{†} *a*^{†}_{l,µ}*a*_{m,µ}*a** _{n,µ}* (4.18)

*where Greek letters refer to AO, h** _{kl}* are the matrix elements of the one-electron

*Hamiltonian and (kn|lm) are the bielectronic repulsion terms. Moreover:*

Φ* _{i→r}* = 1

p2[a^{†}_{1,α}*a*_{1,β}^{†} *. . . a*^{†}_{i,α}*. . . a*^{†}_{r,β}

*−a*^{†}*1,α**a*_{1,β}^{†} *. . . a*^{†}_{i,β}*. . . a*^{†}* _{r,α}*]|0〉 (4.19)
Φ

*= 1*

_{j→s}p2[a^{†}_{1,α}*a*_{1,β}^{†} *. . . a*^{†}_{j,α}*. . . a*^{†}_{s,β}

*−a*^{†}*1,α**a*_{1,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:

*H*^{d}* _{ir, 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 H** ^{d}* in
the RET refer to the non diagonal elements that mix the configurations coming
from the

^{D}excitations with the one from

^{A}. 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 ^{D}and^{A} first at very large distance, to
cal-culate relevant MO, select relevant excitations, and obtain single excited
*con-figuration coefficients C*^{∞}

D*i,r ;*^{A}*j,s*of 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 H*_{ir, js}* ^{d}* 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 2N*C*, in N*C* 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 the^{D}and^{A}separately, obtaining the relevant MO and
exci-tations. Then the AO basis for the supermolecule can be constructed by simply
queuing^{A}AOs after^{D}AOs. The MOs and the CIS matrix for the supermolecule
can be easily rewritten on the composite basis, filling with zeroes. Then the
*calculation of H*_{ir, js}* ^{d}* 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 for*^{D}*and j → s for*^{A}unit
in the way that the configuration transition densities are:

q^{i→r}* _{d}*=X

*δ∈d*

*c*_{iδ}*c** _{rδ}* (4.22)

q^{j→s}* _{a}*=X

*α∈a*

*c*_{jα}*c** _{sα}* (4.23)

where the first equation stand for the^{D} *molecule and δ refers to all the AO*
*centered on the atom d, while the second equation stands for the molecule*^{A}*(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:

˜q* _{d}*D

∗

A

0=X

(i,r)

*C*D*i,r*q^{i→r}* _{d}* (4.24)

˜q* _{a}*D

0

A

∗=X

(j,s)

*C*A*j,s*q^{j→s}* _{a}* (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-interacting^{D}−^{A}pair. The calculation, that involve extensive CIS, can be
conveniently performed separately on^{D}and on^{A}.

In this approximation the RET interaction energy in 4.8 reads:

V=X

*d*

X

*a*

˜q* _{d}*D

∗

A

0˜q* _{a}*D

0

A

∗

1

**|r***d***− r***a*| (4.26)