Effective Electronic Hamiltonian for Quantum Subsystem in Hybrid QM/MM Methods as Derived from APSLG Description of Electronic Structure of Classical Part of Molecular System

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A.M. Tokmachev, A.L. Tchougréeff and I.A. Misurkin
Karpov Institute of Physical Chemistry
Vorontsovo pole 10, Moscow 103064 RUSSIA

Abstract

The general formulae representing separation of electronic variables of quantum (reactive) subsystem from those describing electrons in the classical (chemically inert) part of molecular system are specified for the case when the electronic structure of the latter is described by a semi-empirical method based on the trial wave function having the form of antisymmetrized product of strictly localized geminals (APSLG) which leads to a local description of molecular electronic structure in terms of bond functions and lone pair functions. This allowed us to give explicit form of the effective electronic Hamiltonian for the quantum subsystem and by this also to sequentially derive the explicit form of the QM/MM junction between the quantum and classical subsystems. The latter turned out to be a sum of the contributions from different chemical bonds and lone pairs residing in the classical part of the system. Numerical estimates for the effect of the renormalization of the Coulomb interaction of p-electrons due to presence of s-bonds are performed according to the derived formulae.

Introduction

At present research in chemistry and in related areas of science reached the state that require constructing potential energy surfaces (PES) of large systems. This problem can be encountered in the context of chemical reactions of biomolecules, enzymatic reactions, surface reactions and reactions in condensed media. Applying methods of quantum mechanics (QM) to construct PES's in each point of the nuclear configuration space faces the problem of O(N^m)-scaling of the QM methods (N is the number of one-electron states involved in the calculation). Practically the exponent m may reach values of 5¸7 for high quality modern QM methods necessary to describe chemical reactivity (bond cleavage and bond formation), which restricts their applicability to molecular systems of rather small size []. However, the detailed QM description is necessary for the electronic structure of the reactive site only. The contribution of the rest of molecular system (i.e. of its chemically inert part) to the PES of the system can be calculated by molecular mechanics (MM), which has to reproduce only general features of this part of the molecular system. Thereby the hybrid quantum-mechanical/molecular mechanical (QM/MM) computational schemes become very popular (Refs. Bernardi,Morokuma,Voityuk,Thery,Bersuker,CECAM). This approach significantly reduces the computational costs of PES construction for large systems because only a small part of the latter is considered on the computationally expensive QM level.

There exist several QM/MM schemes implemented in a number of computation packages (see, for example, Refs. Bernardi,Morokuma,Voityuk,Thery,Bersuker,CECAM). The diversity of such approaches is due to variety of both QM and MM methods combined and of the functional form of the junction between them. For example, in Ref. Bernardi the intersystem Coulomb and exchange integrals are represented as linear combinations of exponential functions with subsequent parametrization; in Ref.[] ''junction dummy atoms'' are introduced; in Ref. [] the interactions between subsystems are represented also by a sum of electrostatic and van der Waals interactions with adjusted parameters, etc. The common feature of all the mentioned approaches as well as of many others is that the form of the junction parametrized in each of them results from certain ad hoc postulate rather than from a sequential derivation. On the other hand, in Ref. KhimFiz it was proposed to construct a consistent form of the QM/MM junction with use of an explicit procedure of separation of electronic variables. The latter is performed in Ref. [] by using the Löwdin partition technique [] and the group function (GF) formalism []. In Refs. [,] the expression for the effective Hamiltonian for the quantum (reactive) part of a molecular system has been obtained and the form of the QM/MM junction has been represented as a sum of operator averages over the implicit wave function of the classical (inert) subsystem, which is assumed to describe the electronic ground state of the latter. The present work is devoted to derivation of the explicit form of the effective Hamiltonian for the quantum system and, therefore, of the PES of the combined system and of the specific form of the QM/MM junction for a special local form of the wave function of the inert subsystem which has been constructed in Ref. [] to ensure a ready transition to a description of the MM type which will be decribed elsewhere [].

Effective Hamiltonian for quantum system

Now we briefly review the main notations and results of Ref. [], where the general expression for the effective Hamiltonian has been proposed. We denote two subsystems of the whole molecular system by indici R (reactive) and M (inert), where the R-subsystem is considered as a quantum one, whereas the M-subsystem must be finally treated on the classical (MM) level of approximation. The Hamiltonian for the whole system is a sum of Hamiltonians for the subsystems and of the operators for the interactions between the subsystems:

H = H^R(q)+H^M(q)+V^c(q)+V^r(q),
(0)
where for the sake of simplicity only the Coulomb V^c and the resonance V^r (electron transfer) interactions are considered. Further, the Hamiltonian for the M-subsystem is subdivided into the Hamiltonian for the free (without reagents) M-subsystem H₀^M and the operator describing attraction of electrons of the M-subsystem to the cores of the R-subsystem V^R. Analogous subdividing is performed for the R-subsystem. To justify usage of different levels of approximation to different parts of the whole system (specifically, of the MM-like scheme for the M-subsustem) the wavefunction for the whole system is represented by the antisymmetrized product of electronic wavefunction for the R-subsystem and that of the ground state for the free M-subsystem (i.e. of the ground state function of H₀^M):

Y_k = F_k^RÙF₀₀^M.
(0)
The exact wavefunction of any electronic state of the whole system can be recast in the form:

Y_k =
å
n_Mn_R

å
i_Mi_R
C_{i_Mi_R}^k(n_Mn_R)F_{i_M0}^M(n_M)ÙF_{i_R}^R(n_R), (n_M+n_R = N_e)
(0)
The transition from the wavefunction of the general form Eq.(3) to the necessary form of Eq. (2) is made by performing two sequential Löwdin projection procedures: the first one to the subspace of the states with fixed number of electrons in the subsystems (projection operator P and its complementary projection operator Q = 1-P) and the second one - to the states of the type Eq. 2, i. e. containing the ground state wavefunction of the free M-subsystem as the multiplier (projection operator P and its complementary projection operator Q = 1-P). After the first projection we obtain

H_eff(q,E) = PH^R(q)P+PH^M(q)P+PV^c(q)P+

+PV^rr(q,E)P+ e²
2
å
\Sb A ¹ B

A Î R,B Î M\endSbZ_A^RZ_B^MR_AB^-1,

(0)
where

V^rr(q,E) = V^r(q)Q(E-QHQ)^-1QV^r(q) = V^r(q)QR(q,E)QV^r(q).
(0)
The second projection and subsequent averaging over the ground state of the M-subsystem gives the effective Hamiltonian for the R-subsystem:

H_eff^R(q,E,w) = H₀^R(q)+dV^M+ á áPV^rrP ñ ñ _M+áF₀₀^M | PV^RP | F₀₀^Mñ+

+áF₀₀^M | P W(q,E)PQR(w)QPW(q,E)P | F₀₀^Mñ+ e²
2
å
\Sb A ¹ B

A Î R,B Î M\endSbZ_A^RZ_B^MR_AB^-1,

(0)
where

dV^M = V^M+ á áPV^cP ñ ñ _M » 0
(0)
and

PWP = PV^cP+PV^rrP+PV^RP.
(0)
The above general form of the effective Hamiltonian was obtained in Ref. []. In the present paper we perform the averaging assuming that the wavefunction of the M-subsystem has a specific form, which gives the local description of the latter.

The form of the wave function of the M-subsystem to be used to perform the averaging has to allow to represent the renormalization of Hamiltonian for the quantum subsystem as a sum of contributions from one or more chemical bonds or lone pairs. This is done to maintain consistency with the adopted MM type of description for the M-subsystem. For this purpose we must use a quantum-chemical method, the energy of which can be presented in a MM-like form. The MM scheme assumes the transferability of the functions representing geometry dependence of different contributions to the molecular energy. Such a transferability is shown to be achieved (Ref. []) for the trial wave function in the form of the antisymmetrized product of strictly localized orbitals (APSLG) []. The wave function of this method is constructed from two-electron functions (geminals) assigned to chemical bonds and lone pairs:

| F ñ =
Õ
k
g_k⁺ | 0ñ,
(0)
where

g_k⁺ = u_kr_ka⁺r_kb⁺+v_kl_ka⁺l_kb⁺+w_k(r_ka⁺l_kb⁺+l_ka⁺r_kb⁺),

(u_k²+2w_k²+v_k² = 1)

(0)
is the electronic pair creation operator for the k-th geminal. Each of the orbitals r_k and l_k assigned to the k-th chemical bond is a linear combination of the AO's centered on one atom only, i.e. a hybrid orbital (HO). The unitary matrices of transition from the AO basis to the HO basis and the geminal expansion coefficients u_k, v_k, and w_k are determined variationally in Ref. [9] for a wide range of organic molecules. For the purposes of the present paper it is important to note that the energy of molecule in the APSLG approximation can be represented by a sum of interbond and intrabond (we use the term bond for usual chemical bonds and lone pairs) contributions and that the parameters of these contributions are well transferable. The derivation of the MM description from the QM APSLG method is rather complex and will be published elsewhere [].

Let us consider the averages renormalizing the Hamiltonian for the R-subsystem, carrying out the summation over the spin projections. The operators of the Coulomb and the resonance interactions between the subsystems can be written as

V^c(q) = å
\Sb pp^¢ Î R

mm^¢ Î M\endSb (pp^¢||mm^¢)p⁺m⁺m^¢p^¢,

V^r(q) = å
\Sb p Î R

m Î M\endSb v_pm(q)(p⁺m+m⁺p),

(0)
where

(pp^¢||mm^¢) = (pp^¢ | mm^¢)-(pm^¢ | mp^¢).
(0)
and the indici pp^¢ and mm^¢ refer to the one electron states in the R- and M-subsystems, respectively. In the latter case the one-electron states can be taken as the HO's r_k and l_k in the M-subsystem. The averaging of the operator of Coulomb interaction between the subsystems yields:

á á PV^cP ñ ñ _M = áF₀₀^M| PV^cP| F₀₀^M ñ =

= \stackunderpp^¢ Î R å
p⁺p^¢ é
ë \stackundermm^¢ Î M å
( pp^¢| | mm^¢) á á m⁺m^¢ ñ ñ _M ù
û .

(0)
Since we assume that the M-subsystem is described by the wave function of the APSLG type, all the orbitals in the M-subsystem are either right (r) or left (l) orbitals of geminals. In the APSLG approximation the averages á á m⁺m^¢ ñ ñ _M do not vanish only for the spin-orbitals m and m^¢ belonging to the same geminal and can be written as

á á r_ks⁺r_ks ñ ñ _M = á 0| g_kr_ks⁺r_ksg_k⁺| 0 ñ = P_k^rr = u_k²+w_k²,

á á l_ks⁺l_ks ñ ñ _M = á 0| g_kl_ks⁺l_ksg_k⁺| 0 ñ = P_k^ll = v_k²+w_k²,

á á r_ks⁺l_ks ñ ñ _M = á á l_ks⁺r_ks ñ ñ _M = P_k^rl = (u_k+v_k)w_k.

(0)
It is also convenient to introduce the reduced Coulomb integrals

Y_pp^¢mm^¢ = 2( pp^¢ | mm^¢) -( pm^¢ | mp^¢) .
(0)
Therefore, we can write the average á áPV^cP ñ ñ _M as

á á PV^cP ñ ñ_M =
å
pp^¢ Î R

å
s
p_s ⁺p_s^¢×

×
å
k Î M
[ Y_{pp^¢r_kr_k}P_k^rr+Y_{pp^¢l_kl_k}P_k^ll+(Y_{pp^¢l_kr_k}+Y_{pp^¢r_kl_k})P_k^rl] .

(0)

The ZDO approximation assures that p = p^¢ and m = m^¢. We denote p Î A and m Î B. In the case of A ¹ B we obtain the contributions of the type

å
s
p_s ⁺p_s g_AB é
ë å
\Sb m_k Î B t\endSb á ám_kt⁺m_kt ñ ñ _M ù
û = 2
å
s
p_s ⁺p_s g_AB
å
m_k Î B
P_k^mm.
(0)
The next contribution to dV^M is

V^M = -e²\stackunderB^¢ å
Z_B^¢^M
| r-R_B^¢|
=

= -
å
pp^¢ Î R

å
B Î M
V_B^{pp^¢}Z_B^M
å
s
p_s⁺p_s ^¢.

V_B^{pp^¢} = -e² ó
õ d³r y_p^*(r)y_p^¢(r)
| r-R_B^¢|

(0)
Taking the sum of Eq. (17) and (18) and using the ZDO approximation we obtain that the contribution to dV^M from the interactions of electrons belonging to different atoms equals

å
A Î R

å
p Î AÇR

å
s
p_s ⁺p_s å
\Sb B Î M B ¹ A\endSb g_ABQ_B^M,
(0)
where Q_B^M = 2å\limits_{m_k Î B}P_k^mm-Z_B^M is the effective charge of the atom B. The contribution from the interactions of electrons located on the orbitals belonging to different systems but centered on the same frontier atoms equals to

å
pp^¢ Î A

å
s
p_s ⁺p_s^¢ å
\Sb m_k Î A k Î M\endSb P_k^mmY_pp^¢mm.
(0)
The next contribution to the effective Hamiltonian for the R-subsystem is due to the intersubsystem electron transfers

PV^rr(q,E)P = å
\Sb pp^¢ Î R

mm^¢ Î M\endSb v_pm(q)v_p^¢m^¢(q)×

×[ ( p⁺mR(E)m^¢+p^¢) +(m⁺pR(E)r^¢+p^¢) ] .

(0)
The resolvent can be presented as

R(E) =
å
i Î ImQ
| i ñ á i|
E-E_i
,
(0)
where | i ñ are the states with one electron transferred from the M-subsystem to the R-subsystem and vice versa. We assume that every state | i ñ is an antisymmetrized product of ionized states | m ñ and | r ñ of the M- and R-subsystems respectively. Moreover, we construct the states | m ñ by removing or adding an electron from or to the Dirac orbitals of the M-subsystem. Therefore, we can assume the energy differences in this equation to be expressed through the ionization potentials (IP) and electron affinities (EA):

E_i-E = ì
í
î

I_m -A_r -g_mr

I_r -A_m -g_rm
.
(0)
Now we specify the approximate form of the states | i ñ in the resolvent. First, we notice that the correlations and bonding can be accounted to the same extent as they are in the geminal Eq. (10) if one employs bonding (b) and antibonding (a) bond orbitals (BO) for the k-th geminal (which are also the Dirac orbitals for this geminal):

ì
í
î

b_ks = x_kl_ks+y_kr_ks

a_ks = -y_kl_ks+x_kr_ks
; (x_k²+y_k² = 1).
(0)
for constructing geminals:

g_k⁺ = U_kb_ka⁺b_kb⁺+V_ka_ka⁺a_kb⁺; (U_k²+V_k² = 1).
(0)
The APSLG wave function remains unchanged since the coefficient sets (U_k,V_k;x_k,y_k) and (u_k,w_k,v_k) are uniquely related:

ì
ï
í
ï
î

u_m = U_my_m²+V_mx_m²

v_m = U_mx_m²+V_my_m²

w_m = ( U_m-V_m) x_my_m
.
(0)
Using these bond orbitals we construct the M-multiplier of the charge transfer states | i ñ in the form:

b_ks⁺
Õ
l ¹ k
g_l⁺ | 0ñ,a_ks⁺
Õ
l ¹ k
g_l⁺ | 0ñ,

b_ks⁺r_k-s⁺l_k-s⁺
Õ
l ¹ k
g_l⁺ | 0ñ,a_ks⁺r_k-s⁺l_k-s⁺
Õ
l ¹ k
g_l⁺ | 0ñ.

(0)
The IP's and the EA's for the bond (lone pair) states within the APSLG-MINDO/3 approximation have the form:

I_kb = W_1k^r( y_k²-2P_k^rr) +W_1k^l(x_k²-P_k^ll) +2W_1k^rl( x_ky_k-2P_k^rl) -

-W_2k^ru_k²-W_2k^lv_k²-2W_2k^rlw_k²,

I_ka = W_1k^r( x_k²-2P_k^rr) +W_1k^l(y_k²-P_k^ll) -2W_1k^rl( x_ky_k+2P_k^rl) -

-W_2k^ru_k²-W_2k^lv_k²-2W_2k^rlw_k²,

-A_kb = W_1k^r( 1+y_k²-2P_k^rr) +W_1k^l(1+x_k²-2P_k^ll) +

+2W_1k^rl( x_ky_k-2P_k^rl) +W_2k^r(y_k²-u_k²) +

+W_2k^l( x_k²-v_k²) +2W_2k^rl( 1-w_k²) ,

-A_ka = W_1k^r( 1+x_k²-2P_k^rr) +W_1k^l(1+y_k²-2P_k^ll) -

-2W_1k^rl( x_ky_k+2P_k^rl) +W_2k^r(x_k²-u_k²) +

+W_2k^l( y_k²-v_k²) +2W_2k^rl( 1-w_k²) .

(0)
where these quantities are expressed through the parameters of the effective Hamiltonian for the k-th bond in the APSLG-MINDO/3 approximation, which can be written as

H_k^eff = W_1k^r
å
s
r_ks⁺r_ks+W_1k^l
å
s
l_ks⁺l_ks+W_1k^rl
å
s
( r_ks⁺l_ks+l_ks⁺r_ks) +

+W_2k^rr_ka⁺r_kb⁺r_kbr_ka+W_2k^ll_ka⁺l_kb⁺l_kbl_ka+W_2k^rl
å
s
r_ks⁺l_k-s⁺l_k-sr_ks,

(0)
with

W_1k^r = æ
è U_k^A_k-
å
B^¢ ¹ A_k
g_{A_kB^¢}Z_B^¢ ö
ø +

+
å
t_n Î A_k,n ¹ k
Y_{r_kr_kt_nt_n}^A_kP_n^tt+2
å
B^¢ ¹ A_k
g_{A_kB^¢}
å
t_n Î B^¢,n ¹ k
P_n^tt,

W_1k^l = æ
è U_k^B_k-
å
B^¢ ¹ B_k
g_{B_kB^¢}Z_B^¢ ö
ø +

+
å
t_n Î A_k,n ¹ k
Y_{l_kl_kt_nt_n}^B_kP_n^tt+2
å
B^¢ ¹ B_k
g_{B_kB^¢}
å
t_n Î B^¢,n ¹ k
P_n^tt,

W_1k^rl = -b_k,

W_2k^r = ( r_kr_k | r_kr_k) ^A_k,

W_2k^l = ( l_kl_k | l_kl_k) ^B_k,

W_2k^rl = g_{A_kB_k}.

(0)
The resolvent contribution from the first projection to the effective Hamiltonian of the R-subsystem can be written in the form including the ionized states of the R-subsystem:

á á PV^rr(q,E)P ñ ñ_M = -
å
pp^¢ Î R

å
s

å
k Î M

å
ij Î { r,l}

å
f Î {a,b}
v_{pi_k}v_{p^¢j_k}×

(
å
r Î ImO_R(N_R-1)
p_s ⁺| r ñ á r| p_s ^¢ q_if^kq_jf^k
I_r -A_kf-g_{rf_k}
+

+
å
r Î ImO_R(N_R+1)
p_s | r ñ á r| p_s ^¢+ h_if^kh_jf^k
I_kf-A_r -g_{f_kr}
),

(0)
where we use the vacuum averages

q_if^k = á 0| g_ki_kaf_kb⁺r_ka⁺l_ka⁺| 0 ñ ,

h_if^k = á 0| g_ki_ka⁺f_kb⁺|0 ñ .

(0)
which can be easily expressed in terms of the parameters of the APSLG wave function in the representations Eq. (10) and Eqs. (25), (26):

q_rb^k = y_kw_k+x_kv_k, q_ra^k = x_kw_k-y_kv_k,

q_lb^k = -x_kw_k-y_ku_k, q_la^k = y_kw_k-x_ku_k;

h_rb^k = y_ku_k+x_kw_k, h_ra^k = x_ku_k-y_kw_k,

h_lb^k = y_kw_k+x_kv_k, h_la^k = x_kw_k-y_kv_k.

(0)

Now we consider rather cumbersome contribution which arise during the second projection:

áF₀₀^M | PW(q,E)PQR(w)QPW(q,E)P | F₀₀^Mñ.
(0)
In order to do that we reconsider the substance of the notion of the quantum character of the R-subsystem and of the classical one of the M-subsystem. As it is mentioned in Ref. [] the quantum character of a part of a molecular system manifests itself in its spectrum which posesses excited states in a narrow energy range close to its ground state. It makes it possible to observe several quantum states in experiment at least in principle. By contrast the characteristic of a classical part of molecular system is that its properties are determined by its ground state only so that the energies of its excited states are very high as compared to the energy range probed experimentaly. In the present derivation we are interested in obtaining the effective Hamiltonian for the states of the R- (quantum) subsystem close to its ground state. We assume that the dependence of the resolvent on w is weak and the w values of interest are much smaller than the resolvent poles which are all higher that the first extitation energy in the M- (classical) subsystem which in its turn is higher than the excitation energies of interest in the R-subsystem. Therefore we can turn to the limit w® 0 in the expression (34). The resolvent can be represented as

Â(0) =
lim
w® 0
Â(w) =
lim
w® 0
å
\Sb rm ¹ 0\endSb | r,m ñ á r,m|
w-w_rm
= - å
\Sb rm ¹ 0\endSb | r,m ñ á r,m|
e_r +e_m
,
(0)
where the sum excludes the states having the ground state of the M-subsystem as the multiplier and e_r and e_m are the energies of the excitations in the R- and M-subsystems, respectively. The contribution Eq.(34) can be expressed as:

áF₀₀^M | P W(q,E)PQR(0)QPW(q,E)P | F₀₀^Mñ =

= áF₀₀^M | PV^cP QR(0)QPV^cP | F₀₀^Mñ+

+áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ+

+áF₀₀^M | P V^RPQR(0)QPV^RP | F₀₀^Mñ+

+2áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ+

+2áF₀₀^M | PV^cP QR(0)QPV^RP | F₀₀^Mñ+

+2áF₀₀^M | PV^rrPQR(0)QPV^RP | F₀₀^Mñ.

(0)
We will write these averages explicitly. Since the R-subsystem is quantum and thus has low-lying excited states but the M-subsystem is classical and its excitation energies are high and thus e_r << e_m and e_r can be neglected in Eq. (35) as compared to e_m . Using the symmetry with respect to the spin indici we obtain:

áF₀₀^M | PV^cP QR(0)QPV^cP | F₀₀^Mñ =

=
å
pp^¢qq^¢ Î R

å
kn Î M

å
ii^¢jj^¢ Î { r,l}
[ å
\Sb st

s^¢t^¢\endSb p_s ⁺p_s ^¢( 1-| 0_R ñ á 0_R| ) q_t⁺q_t ^¢P_{i_ks^¢i_ns^¢^¢j_nt^¢j_kt^¢^¢}^M×

×((pp^¢ | i_ki_n^¢)-d_ss^¢(pi_n^¢ | i_kp^¢))((qq^¢ | j_nj_k^¢)-d_tt^¢(qj_k^¢ | j_nq^¢))+

+
å
s
p_s ⁺p_-s^¢( 1-|0_R ñ á 0_R| ) q_-s⁺q_s^¢(pi_n^¢ | i_kp^¢)(qj_k^¢ | j_nq^¢)P_{i_k-si_ns^¢j_nsj_k-s^¢}^M],

(0)
where the state | 0_R ñ is the ground state of the free R -subsystem. In this expression we use the zero frequency polarization propagators of the M-subsystem [], which are defined by the expression

P_mm^¢nn^¢^M = -
å
m ¹ 0
á F₀₀^M| m⁺m^¢| m ñ á m| n⁺n^¢| F₀₀^M ñ
e_m
.
(0)
to which the excitations in the M-subsystem contribute. The APSLG form of the ground state wave function implies specific classification for these excitations: they are either intrabond singlet - singlet or singlet - triplet excitations or the interbond one- and two-electron transfers. We present the explicit expressions for the contributions to the polarization propagator from the excitations of different types.

1. P_{i_ksi_ks^¢j_ktj_kt^¢}^M . The excited state | m ñ is one of the states | m_1,2 ñ = g_k^(1,2)+Õ\limits_{k^¢ ¹ k}g_k^¢⁺| 0 ñ (these are excited singlet configurations of the k-th geminal with others unchanged) and | m₃ ñ = [1/(Ö2)](r_ka⁺l_kb⁺-l_ka⁺r_kb⁺)Õ\limits_{k^¢ ¹ k}g_k^¢⁺| 0 ñ (the triplet configuration with s_z = 0 of the k-th geminal with others unchanged). Therefore,

P_{i_ksi_ks^¢j_ktj_kt^¢}^M = -
å
s Î { 1¸3}
á F₀₀^M| i_ks⁺i_ks^¢| m_s ñ á m_s| j_kt⁺j_kt^¢| F₀₀^M ñ
e_{m_s}
.
(0)
The energies of excitations e_{m_s} in the APSLG-MINDO/3 approximation Ref. [9] are:

e_{m_{s = 1,2}} = 2W_1k^r(u_k^(s)2+w_k^(s)2-P_k^rr) +

+2W_1k^l( v_k^(s)2+w_k^(s)2-P_k^ll) +

+W_2k^r( u_k^(s)2-u_k²) +W_2k^l(v_k^(s)2-v_k²) +

+4W_1k^rl[ ( u_k^(s)+v_k^(s))w_k^(s)-P_k^rl] +2W_2k^rl( w_k^(s)2-w_k²) ,

(0)

e_m₃ = ( W_1k^r-W_1k^l) ( v_k²-u_k²)-

-W_2k^ru_k²-W_2k^lv_k²-4W_1k^rlP_k^rl+W_2k^rl(u_k²+v_k²) .

(0)
The matrix elements entering the expression for the polarization propagator are

á F₀₀^M| r_ks⁺r_ks| m_{s = 1,2} ñ = u_ku_k^(s)+w_kw_k^(s),

á F₀₀^M| l_ks⁺l_ks| m_{s = 1,2} ñ = v_kv_k^(s)+w_kw_k^(s),

á F₀₀^M| r_ks⁺l_ks| m_{s = 1,2} ñ = u_kw_k^(s)+w_kv_k^(s),

á F₀₀^M| l_ks⁺r_ks| m_{s = 1,2} ñ = w_ku_k^(s)+v_kw_k^(s);

(0)

á F₀₀^M| r_ka⁺r_ka| m₃ ñ = w_k
Ö2
, á F₀₀^M| l_ka⁺l_ka| m₃ ñ = - w_k
Ö2
,

á F₀₀^M| r_ka⁺l_ka| m₃ ñ = -u_k
Ö2
, á F₀₀^M| l_ka⁺r_ka| m₃ ñ = v_k
Ö2

á F₀₀^M| i_kb⁺j_kb| m₃ ñ = - á F₀₀^M| i_ka⁺j_ka| m₃ ñ .

(0)
For real orbitals the following á m_s| i⁺j|F₀₀^M ñ = á F₀₀^M| j⁺i| m_s ñ holds.

2. P_{i_k-si_ks^¢j_ksj_k-s^¢}^M. The excited state | m₄ ñ = r_ks⁺l_ks⁺Õ\limits_{k^¢ ¹ k}g_k^¢⁺| 0 ñ (the triplet configuration with s_z = 1 for s = a and s_z = -1 for s = b in the k-th geminal with others unchanged).

á F₀₀^M| i_k-s⁺i_ks^¢| m₄ ñ = á 0| g_ki_k-s⁺i_ks^¢r_ks⁺l_ks⁺|0 ñ .
(0)
The energies of such excitations obviously equal to respective energies of excitations to the triplet states with s_z = 0, i.e. e_m₄ = e_m₃. The matrix elements equal to

á F₀₀^M| r_ka⁺r_kb| m₄ ñ = w_k, á F₀₀^M| l_ka⁺l_kb| m₄ ñ = -w_k,

á F₀₀^M| l_ka⁺r_kb| m₄ ñ = v_k, á F₀₀^M| r_ka⁺l_kb| m₄ ñ = -u_k

á F₀₀^M| i_kb⁺j_ka| m₄ ñ = - á F₀₀^M| i_ka⁺j_kb| m₄ ñ .

(0)

3. P_{i_ksi_ns^¢j_ntj_kt^¢}^M(n ¹ k). This polarization propagator differs from zero only for s = t. The excited state involved is | m₅ ñ = h_n-s⁺r_ns⁺l_ns⁺f_k-s⁺Õ\limits_{k^¢ ¹ k,n}g_k^¢⁺|0 ñ , where f and h are either b or a BO's. The energy of the excitation is estimated as

e₅ = I_kf-A_nh-g_{f_kh_n}.
(0)
The IP's and the EA's (I_kf,A_nh) are given above. Therefore, we specify here only g_{f_kh_l}. It is convenient to introduce new quantities c_fi, which are amplitudes of the i-th HO (r or l) in the BO f (a or b) given by Eq. (24). Then

g_{f_kh_n} =
å
ij Î { r,l}
[Y_{i_ki_kj_nj_n}(2P_kⁱⁱ( 1+c_{f_ki_k}²) +2P_n^jjc_{h_nj_n}²) -

-( i_ki_k | j_nj_n) ( 1+c_{f_ki_k}²)c_{h_nj_n}²+( i_kj_n | i_kj_n) c_{f_ki_k}²c_{h_nj_n}²].

(0)
The required matrix elements are

á F₀₀^M| i_ks⁺i_ns^¢| m₅ ñ = h_if^kq_i^¢hⁿ.
(0)

4. P_{i_k-si_ns^¢j_nsj_k-s^¢}^M(n ¹ k). The excited state is | m₆ ñ = h_n-s⁺r_ns⁺l_ns⁺f_ks⁺Õ\limits_{k^¢ ¹ k,n}g_k^¢⁺|0 ñ , where f and h are either b or a. The energy of the excitation is estimated as

e₆ = I_kf-A_nh-g_{f_kh_n}^¢,
(0)
where

g_{f_kh_n}^¢ =
å
ij Î { r,l}
[Y_{i_ki_kj_nj_n}( 2P_kⁱⁱ( 1+c_{f_ki_k}²)+2P_n^jjc_{h_nj_n}²) -

-( i_ki_k | j_nj_n) ( 1+c_{f_ki_k}²)c_{h_nj_n}²+( i_kj_n | i_kj_n) c_{h_nj_n}²].

(0)
The required matrix elements are

á F₀₀^M| i_k-s⁺i_ns^¢| m₆ ñ = -h_if^kq_i^¢hⁿ.
(0)
If the ZDO approximation is used for the two-center Coulomb interaction parameters the contribution Eq. (34) into the effective Hamiltonian for the R-subsystem simplifies significantly:

áF₀₀^M | PV^cP QR(0)QPV^cP | F₀₀^Mñ =

=
å
pq Î R

å
k Î M

å
ij Î { r,l}
[ å
\Sb st

s^¢t^¢\endSb p_s ⁺p_s ( 1-|0_R ñ á 0_R| ) q_t ⁺q_t P_{i_ks^¢i_ks^¢j_kt^¢j_kt^¢}^M×

×((pp | i_ki_k)-d_ss^¢(pi_k | i_kp))((qq | j_kj_k)-d_tt^¢(qj_k | j_kq))+

+
å
s
p_s ⁺p_-s( 1-|0_R ñ á 0_R| ) q_-s⁺q_s(pi_k | i_kp)(qj_k | j_kq)P_{i_k-si_ksj_ksj_k-s}^M].

(0)
The next contribution to the effective Hamiltonian can be presented as a sum of one-geminal and two-geminal contributions:

áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ =

å
k
áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ_k+

+
å
k ¹ n
áF₀₀^M | PV^rrPQR(0)QPV^rrP | F₀₀^Mñ_kn.

(0)
The one-geminal contribution has the form

áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ_k =

= å
\Sbpp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

fh Î { a,b} \endSb å
\Sb st

V V ^¢\endSb v_{pi_k}v_{p^¢i_k^¢}v_{qj_k}v_{q^¢j_k^¢}×

×[
å
r₁,r₂ Î ImO_R(N_R+1)
A_pp^¢qq^¢^{stV V ^¢}( r₁r₂) X_{i_ksi_kV^¢j_{kV ^¢}j_kt^¢}^f_k-sh_k-t
(I_kf-A_r₁-g_{f_kr₁})(I_kh-A_r₂-g_{h_kr₂})
-

- å
\Sb r₁ Î ImO_R(N_R+1)

r₂ Î ImO_R(N_R-1)\endSb ( -1) ^{d_jj^¢} B_pp^¢qq^¢^{stV V ^¢}( r₁r₂) X_{i_ksi_kV^¢ [(j)\tilde]_{kV ^¢}[(j)\tilde]_kt^¢}^f_k-sh_k-t
(I_kf-A_r₁-g_{f_kr₁})(I_r₂-A_kh-g_{r₂h_k})
-

- å
\Sb r₁ Î ImO_R(N_R-1)

r₂ Î ImO_R(N_R+1)\endSb ( -1) ^{d_ii^¢} C_pp^¢qq^¢^{stV V ^¢}( r₁r₂) X_{[(i)\tilde]_ks[(i)\tilde]_kV^¢j_{kV ^¢}j_kt^¢}^f_k-sh_k-t
(I_r₁-A_kf-g_{r₁f_k})(I_kh-A_r₂-g_{h_kr₂})
+

+
å
r₁,r₂ Î ImO_R(N_R-1)
(-1)^{d_ii^¢+d_jj^¢} D_pp^¢qq^¢^{stV V ^¢}( r₁r₂) X_{[(i)\tilde]_ks[(i)\tilde]_kV^¢[(j)\tilde]_{kV ^¢}[(j)\tilde]_kt^¢}^f_k-sh_k-t
(I_r₁-A_kf-g_{r₁f_k})(I_r₂-A_kh-g_{r₂h_k})
],

(0)
while the two-geminal contributions have the form

áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ_kn =

= å
\Sb pp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

f Î { a,b} \endSb
å
sV
v_{pi_k}v_{p^¢i_n^¢}v_{qj_n}v_{q^¢j_k^¢}×

×[
å
r₁,r₂ Î ImO_R(N_R+1)
A_pp^¢qq^¢^{ssV V}( r₁r₂) X_{i_ksi_nV^¢j_nVj_ks^¢}^f_k-sf_k-s
(I_kf-A_r₁-g_{f_kr₁})(I_kf-A_r₂-g_{f_kr₂})
+

+
å
r₁,r₂ Î ImO_R(N_R-1)
D_pp^¢qq^¢^{ssV V}( r₁r₂) X_{i_nV^¢i_ksj_ks^¢j_nV}^f_k-sf_k-s
(I_r₁-A_kf-g_{r₁f_k})(I_r₂-A_kf-g_{r₂f_k})
]-

+ å
\Sb pp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

fh Î { a,b} \endSb
å
sV
v_{pi_k}v_{p^¢i_n^¢}v_{qj_k}v_{q^¢j_n^¢}(-1)^d_sV×

×[ å
\Sb r₁ Î ImO_R(N_R+1)

r₂ Î ImO_R(N_R-1)\endSb B_pp^¢qq^¢^{ssV V}( r₁r₂) X_{i_ksi_nV^¢j_nV^¢j_ks}^f_k-sh_n-V
(I_kf-A_r₁-g_{f_kr₁})(I_r₂-A_nh-g_{r₂h_n})
+

+ å
\Sb r₁ Î ImO_R(N_R-1)

r₂ Î ImO_R(N_R+1)\endSb C_pp^¢qq^¢^{ssV V}( r₁r₂) X_{i_nV^¢i_ksj_ksj_nV^¢}^f_k-sh_n-V
(I_r₁-A_kf-g_{r₁f_k})(I_nh-A_r₂-g_{h_nr₂})
],

(0)
where [(i)\tilde] = r for i = l and [(i)\tilde] = l for i = r. Operators can be represented as

A_pp^¢qq^¢^{stV V ^¢}( r₁r₂) = d_{V s}d_{V ^¢t}p_s | r₁ ñ á r₁| p_s ^¢+( 1-|0_R ñ á 0_R| ) q_t | r₂ ñ á r₂| q_t ^¢++

+d_{V V ^¢}d_std_{V -s}p_s | r₁ ñ á r₁| p_-s^¢+( 1-| 0_R ñ á 0_R| ) q_-s| r₂ ñ á r₂| q_s ^¢+,

B_pp^¢qq^¢^{stV V ^¢}( r₁r₂) = d_{V s}d_{V ^¢t}p_s | r₁ ñ á r₁| p_s ^¢+( 1-|0_R ñ á 0_R| ) q_t ⁺| r₂ ñ á r₂| q_t ^¢-

-d_{V V ^¢}d_std_{V -s}p_s | r₁ ñ á r₁| p_-s^¢+( 1-| 0_R ñ á 0_R| ) q_-s⁺| r₂ ñ á r₂| q_s ^¢,

C_pp^¢qq^¢^{stV V ^¢}( r₁r₂) = d_{V s}d_{V ^¢t}p_s ⁺| r₁ ñ á r₁| p_s ^¢( 1-|0_R ñ á 0_R| ) q_t | r₂ ñ á r₂| q_t ^¢+-

-d_{V V ^¢}d_std_{V -s}p_s ⁺| r₁ ñ ár₁| p_-s^¢( 1-| 0_R ñ á 0_R| ) q_-s| r₂ ñ á r₂| q_s ^¢+,

D_pp^¢qq^¢^{stV V ^¢}( r₁r₂) = d_{V s}d_{V ^¢t}p_s ⁺| r₁ ñ á r₁| p_s ^¢( 1-|0_R ñ á 0_R| ) q_t ⁺| r₂ ñ á r₂| q_t ^¢+

+d_{V V ^¢}d_std_{V -s}p_s ⁺| r₁ ñ ár₁| p_-s^¢( 1-| 0_R ñ á 0_R| ) q_-s⁺| r₂ ñ á r₂| q_s ^¢.

(0)
Also we introduce the quantities

X_ii^¢jj^¢^fh = -
å
m ¹ 0
á F₀₀^M| i⁺f⁺fi^¢|m ñ á m| j⁺h⁺hj^¢| F₀₀^M ñ
e_m
,
(0)
which in general case can not be reduced to the polarization propagators. Now we consider different cases (The energies of excitations were determined above).

1. X_{i_ksi_ks^¢j_ksj_ks^¢}^f_k-sh_k-s. The excited state | m ñ is one of | m_1,2,3 ñ , defined above. We have to determine only matrix elements.

á F₀₀^M| i_ks⁺f_k-s⁺f_k-si_ks^¢| m_{s = 1,2} ñ = q_if^kq_i^¢f^k(s),

(0)
where q_if^k(s) coincides with q_if^k defined abobe (see Eq. (32)), but with u_k, v_k, and w_k changed to u_k^(s), v_k^(s), and w_k^(s).

á F₀₀^M| i_ks⁺f_k-s⁺f_k-si_ks^¢| m₃ ñ = 1
Ö2
q_if^k( d_i^¢rc_{f_kl_k}-d_i^¢lc_{f_kr_k}) .
(0)

2. X_{i_k-si_ks^¢j_ksj_k-s^¢}^f_ksh_ks. The excited state is | m₄ ñ . The matrix elements are

á F₀₀^M| i_k-s⁺f_ks⁺f_ksi_ks^¢| m₄ ñ = (-1)^d_saq_if^k( d_i^¢rc_{f_kl_k}-d_i^¢lc_{f_kr_k}) .
(0)

3. X_{i_ksi_ns^¢j_nsj_ks^¢}^f_k-sf_k-s. The excited state is of | m₅ ñ type. The required matrix elements are

á F₀₀^M| i_ks⁺f_k-s⁺f_k-si_ns^¢| m₅ ñ = h_if^kq_i^¢hⁿ.
(0)

4. X_{i_ns^¢i_ksj_ks^¢j_ns}^f_k-sf_k-s. The excited state is of | m₅ ñ type: f_k-s⁺r_ks⁺l_ks⁺h_n-s⁺Õ\limits_{k^¢ ¹ k,n}g_k^¢⁺| 0 ñ . Therequired matrix elements are

á F₀₀^M| i_ns^¢+f_k-s⁺f_k-si_ks| m₅ ñ = q_if^kh_i^¢fⁿ.
(0)

5. X_{i_k-si_ns^¢j_nsj_k-s^¢}^f_ksf_ks. The excited state is of |m₆ ñ type. The required matrix elements are

á F₀₀^M| i_k-s⁺f_ks⁺f_ksi_ns^¢| m₆ ñ = -h_if^kq_i^¢hⁿ.
(0)

6. X_{i_ns^¢i_k-sj_k-s^¢j_ns}^f_ksf_ks. The excited state is of |m₆ ñ type: f_ks⁺r_k-s⁺l_k-s⁺h_n-s⁺Õ\limits_{k^¢ ¹ k,n}g_k^¢⁺| 0 ñ . The required matrix elements are

á F₀₀^M| i_ns^¢+f_ks⁺f_ksi_k-s| m₆ ñ = -q_if^kh_i^¢hⁿ.
(0)
Other quantities X do not contain new averages and can be obtained using expressions written above.

Also we can note that

P_{i_ksi_ks^¢j_ksj_ks^¢}^M =
å
fh Î { r,l}
X_{i_ksi_ks^¢j_ksj_ks^¢}^f_k-sh_k-s
(0)
and analogous expressions can be obtained for other types of X, i.e., the summation over upper indici of X's gives the polarization propagator with respective lower indici. Therefore, if we use the approximation that the energies of electron transfer from (or to) the k-th bond of the M-subsystem to (or from) R-subsystem do not depend on the type of the resultant state of the bond (or equivaqlently consider only one excited state of each type for ) we obtain this contribution to the effective Hamiltonian for the R-subsystem to be expressed in terms of the polarization propagators:

áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ_k =

= å
\Sbpp^¢ Î R

qq^¢ Î R\endSb
å
ii^¢jj^¢ Î {r,l}
å
\Sb st

V V ^¢\endSb v_{pi_k}v_{p^¢i_k^¢}v_{qj_k}v_{q^¢j_k^¢}×

×[
å
r₁,r₂ Î ImO_R(N_R+1)
A_pp^¢qq^¢^{stV V ^¢}( r₁r₂) P_{i_ksi_kV^¢j_{kV ^¢}j_kt^¢}^M
(I_k-A_r₁-g_kr₁)(I_k-A_r₂-g_kr₂)
-

- å
\Sb r₁ Î ImO_R(N_R+1)

r₂ Î ImO_R(N_R-1)\endSb ( -1) ^{d_jj^¢} B_pp^¢qq^¢^{stV V ^¢}( r₁r₂) P_{i_ksi_kV^¢ [(j)\tilde]_{kV ^¢}[(j)\tilde]_kt^¢}^M
(I_k-A_r₁-g_kr₁)(I_r₂-A_k-g_r₂k)
-

- å
\Sb r₁ Î ImO_R(N_R-1)

r₂ Î ImO_R(N_R+1)\endSb ( -1) ^{d_ii^¢} C_pp^¢qq^¢^{stV V ^¢}( r₁r₂) P_{[(i)\tilde]_ks[(i)\tilde]_kV^¢j_{kV ^¢}j_kt^¢}^M
(I_r₁-A_k-g_r₁k)(I_k-A_r₂-g_kr₂)
+

+
å
r₁,r₂ Î ImO_R(N_R-1)
(-1)^{d_ii^¢+d_jj^¢} D_pp^¢qq^¢^{stV V ^¢}( r₁r₂) P_{[(i)\tilde]_ks[(i)\tilde]_kV^¢[(j)\tilde]_kV^¢[(j)\tilde]_kt^¢}^M
(I_r₁-A_k-g_r₁k)(I_r₂-A_k-g_r₂k)
],

(0)
and

áF₀₀^M | PV^rrP QR(0)QPV^rrP | F₀₀^Mñ_kn =

= å
\Sb pp^¢ Î R

qq^¢ Î R\endSb
å
ii^¢jj^¢ Î {r,l}

å
sV
v_{pi_k}v_{p^¢i_n^¢}v_{qj_n}v_{q^¢j_k^¢}×

×[
å
r₁,r₂ Î ImO_R(N_R+1)
A_pp^¢qq^¢^{ssV V}( r₁r₂) P_{i_ksi_nV^¢j_nVj_ks^¢}^M
(I_k-A_r₁-g_kr₁)(I_k-A_r₂-g_kr₂)
+

+
å
r₁,r₂ Î ImO_R(N_R-1)
D_pp^¢qq^¢^{ssV V}( r₁r₂) P_{i_nV^¢i_ksj_ks^¢j_nV}^M
(I_r₁-A_k-g_r₁k)(I_r₂-A_k-g_r₂k)
]-

+ å
\Sb pp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

fh Î { a,b} \endSb
å
sV
v_{pi_k}v_{p^¢i_n^¢}v_{qj_k}v_{q^¢j_n^¢}(-1)^d_sV×

×[ å
\Sb r₁ Î ImO_R(N_R+1)

r₂ Î ImO_R(N_R-1)\endSb B_pp^¢qq^¢^{ssV V}( r₁r₂) P_{i_ksi_nV^¢j_nV^¢j_ks}^M
(I_k-A_r₁-g_kr₁)(I_r₂-A_n-g_r₂n)
+

+ å
\Sb r₁ Î ImO_R(N_R-1)

r₂ Î ImO_R(N_R+1)\endSb C_pp^¢qq^¢^{ssV V}( r₁r₂) P_{i_nV^¢i_ksj_ksj_nV^¢}^M
(I_r₁-A_k-g_r₁k)(I_n-A_r₂-g_nr₂)
],

(0)
The next contribution to the effective Hamiltonian for the R-subsystem is a c-number. The operator V^R can be written as

V^R = -
å
mm^¢ Î M

å
s
m_s⁺m_s ^¢
å
B Î R
V_B^{mm^¢}Z_B^R

» - å
\Sb A Î M\endSb
å
m Î AÇM

å
s
m_s ⁺m_s å
\Sb B Î R

B ¹ A\endSb g_ABZ_B^R

(0)
Respective contribution is

áF₀₀^M | P V^RPQR(0)QPV^RP | F₀₀^Mñ =

=
å
st

å
kn Î M

å
ii^¢jj^¢ Î { r,l}

å
BB^¢ Î R
V_B^{i_ki_n^¢}V_B^¢^{j_nj_k^¢}Z_B^RZ_B^¢^RP_{i_ksi_ns^¢j_ntj_kt^¢}^M »

»
å
st

å
k Î M
P_{r_ksr_ksr_ktr_kt}^M
å
BB^¢ Î R
Z_B^RZ_B^¢^R×

×( (1-d_{BA_k})g_{BA_k}-(1-d_{BB_k})g_{BB_k}) ×

×( (1-d_{B^¢A_k})g_{B^¢A_k}-(1-d_{B^¢B_k})g_{B^¢B_k}) .

(0)
From this approximate expression we can portion out the contribution which includes BB^¢ Ï { A_k,B_k} :

å
st

å
k Î M
P_{r_ksr_ksr_ktr_kt}^M å
\Sb BB^¢ Î R BB^¢ Ï { A_k,B_k} \endSbZ_B^RZ_B^¢^R( g_{BA_k}-g_{BB_k}) (g_{B^¢A_k}-g_{B^¢B_k}) .
(0)
This contribution can be expressed through the bond polarizabilities. The two center integrals g_AB can be approximated by their values calculated at the center of the k-th bond with the corrections linear in interatomic vectors:

g_{BA_k} = e(V(R_B^(k))- 1
2
(ÑV(R_B^(k))R_{A_kB_k})),
(0)
where V(R_B^(k)) is the potential induced by a unit charge placed on the atom B at the center of the k-th bond; the correction contains the gradient of this potential. Substituting these expressions into Eq. ( 70) we get:

å
st
å
\Sb BB^¢ Î R

BB^¢ Ï { A_k,B_k} \endSb Z_B^RZ_B^¢^R
å
k Î M
(ÑV(R_B^(k))m_{A_kB_k}^(k))×

×(ÑV(R_B^¢^(k))m_{A_kB_k}^(k))P_{r_ksr_ksr_ktr_kt}^M =

= - å
\Sb BB^¢ Î R

BB^¢ Ï { A_k,B_k} \endSb Z_B^RZ_B^¢^R
å
k Î M
(ÑV(R_B^¢^(k)) ê
ê ^
a

(k)

(0) ê
ê ÑV(R_B^(k))),

(0)
where the standard expressions reviewed in Ref. [] for the bond polarizability tensors [^(a)] ^(k)(w) for the k-th bond through the polarization propagator for the corresponding geminal and the bond dipole vector m_{A_kB_k}^(k) of the k-th bond. The bond polarizabilities [^(a)] ^(k)(0) are tabulated, for example, in Ref. [].

The cross term between the PV^cP and PV^rrP is a sum of one-geminal and two-geminal contributions:

áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ =

å
k
áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ_k+

+
å
k ¹ n
áF₀₀^M | PV^cPQR(0)QPV^rrP | F₀₀^Mñ_kn.

(0)
The one-geminal contribution equals to

áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ_k = å
\Sbpp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î { r,l}

f Î { a,b} \endSb
å
ss^¢t
v_{qj_k}v_{q^¢j_k^¢}×

×[
å
r₁ Î ImO_R(N_R+1)
F_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_k^¢)-d_ss^¢(pi_k^¢ | i_kp^¢))W_{i_ks^¢i_ks^¢^¢j_ktj_kt^¢}^f_k-t
(I_kf-A_r₁-g_{f_kr₁})
-

-
å
r₁ Î ImO_R(N_R+1)
d_ss^¢d_stF_pp^¢qq^¢^s-s-ss( r₁)(pi_k^¢ | i_kp^¢)W_{i_k-si_ks^¢j_ksj_k-s^¢}^f_ks
(I_kf-A_r₁-g_{f_kr₁})
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} G_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_k^¢)-d_ss^¢(pi_k^¢ | i_kp^¢))W_{i_ks^¢i_ks^¢^¢[(j)\tilde]_kt[(j)\tilde]_kt^¢}^f_k-t
(I_r₁-A_kf-g_{r₁f_k})
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} d_ss^¢d_stG_pp^¢qq^¢^s-s-ss( r₁)(pi_k^¢ | i_kp^¢)W_{i_k-si_ks^¢[(j)\tilde]_ks[(j)\tilde]_k-s^¢}^f_ks
(I_r₁-A_kf-g_{r₁f_k})
],

(0)
while the two-geminal contribution is

áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ_kn =

= å
\Sb pp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

f Î { a,b} \endSb
å
st

å
r₁ Î ImO_R(N_R+1)
v_{qj_n}v_{q^¢j_k^¢}×

×[ F_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_n^¢)-d_st(pi_n^¢ | i_kp^¢))W_{i_kti_nt^¢j_ntj_kt^¢}^f_k-t
(I_kf-A_r₁-g_{f_kr₁})
-

- d_stF_pp^¢qq^¢^s-s-ss( r₁) (pi_n^¢ | i_kp^¢)W_{i_k-si_ns^¢j_nsj_k-s^¢}^f_ks
(I_kf-A_r₁-g_{f_kr₁})
]-

- å
\Sb pp^¢ Î R

qq^¢ Î R\endSb å
\Sb ii^¢jj^¢ Î {r,l}

f Î { a,b} \endSb
å
st

å
r₁ Î ImO_R(N_R-1)
v_{qj_k}v_{q^¢j_n^¢}×

×[ G_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_n^¢)-d_st(pi_n^¢ | i_kp^¢))W_{i_kti_nt^¢j_nt^¢j_kt}^f_n-t
(I_r₁-A_nf-g_{r₁f_n})
-

- d_stG_pp^¢qq^¢^s-s-ss( r₁) (pi_n^¢ | i_kp^¢)W_{i_k-si_ns^¢j_ns^¢j_k-s}^f_n-s
(I_r₁-A_nf-g_{r₁f_n})
],

(0)
where the operator multipliers are:

F_pp^¢qq^¢^{ss^¢tt^¢}( r₁) = p_s ⁺p_s^¢^¢(1-| 0_R ñ á 0_R| ) q_t | r₁ ñ á r₁| q_t^¢^¢+,

G_pp^¢qq^¢^{ss^¢tt^¢}( r₁) = p_s ⁺p_s^¢^¢(1-| 0_R ñ á 0_R| ) q_t ⁺|r₁ ñ á r₁| q_t^¢^¢,

(0)
and new quantity which is somewhat intermediate between P and X are introduced:

W_ii^¢jj^¢^f = -
å
m ¹ 0
á F₀₀^M| i⁺i^¢| m ñ á m| j⁺f⁺fj^¢| F₀₀^M ñ
e_m
.
(0)
All the excited states, their relative energies and matrix averages are determined above. As for the X's, the summation over upper indici of the W's gives the polarization propagator. Therefore, assuming that ionization energies for the different BO's of the k-th geminal are equal we obtain

áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ_k = å
\Sbpp^¢ Î R

qq^¢ Î R\endSb
å
ii^¢jj^¢ Î { r,l}

å
ss^¢t
v_{qj_k}v_{q^¢j_k^¢}×

×[
å
r₁ Î ImO_R(N_R+1)
F_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_k^¢)-d_ss^¢(pi_k^¢ | i_kp^¢))P_{i_ks^¢i_ks^¢^¢j_ktj_kt^¢}^M
(I_k-A_r₁-g_kr₁)
-

-
å
r₁ Î ImO_R(N_R+1)
d_ss^¢d_stF_pp^¢qq^¢^s-s-ss( r₁)(pi_k^¢ | i_kp^¢)P_{i_k-si_ks^¢j_ksj_k-s^¢}^M
(I_k-A_r₁-g_kr₁)
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} G_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_k^¢)-d_ss^¢(pi_k^¢ | i_kp^¢))P_{i_ks^¢i_ks^¢^¢[(j)\tilde]_kt[(j)\tilde]_kt^¢}^M
(I_r₁-A_k-g_r₁k)
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} d_ss^¢d_stG_pp^¢qq^¢^s-s-ss( r₁) (pi_k^¢ | i_kp^¢)P_{i_k-si_ks^¢[(j)\tilde]_ks[(j)\tilde]_k-s^¢}^M
(I_r₁-A_k-g_r₁k)
],

(0)
for the one-geminal contributions and

áF₀₀^M | PV^cP QR(0)QPV^rrP | F₀₀^Mñ_kn =

= å
\Sb pp^¢ Î R

qq^¢ Î R\endSb
å
ii^¢jj^¢ Î {r,l}

å
st

å
r₁ Î ImO_R(N_R+1)
v_{qj_n}v_{q^¢j_k^¢}×

×[ F_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_n^¢)-d_st(pi_n^¢ | i_kp^¢))P_{i_kti_nt^¢j_ntj_kt^¢}^M
(I_k-A_r₁-g_kr₁)
-

- d_stF_pp^¢qq^¢^s-s-ss( r₁) (pi_n^¢ | i_kp^¢)P_{i_k-si_ns^¢j_nsj_k-s^¢}^M
(I_k-A_r₁-g_kr₁)
]-

- å
\Sb pp^¢ Î R

qq^¢ Î R\endSb
å
ii^¢jj^¢ Î {r,l}

å
st

å
r₁ Î ImO_R(N_R-1)
v_{qj_k}v_{q^¢j_n^¢}×

×[ G_pp^¢qq^¢^sstt( r₁) ((pp^¢ | i_ki_n^¢)-d_st(pi_n^¢ | i_kp^¢))P_{i_kti_nt^¢j_nt^¢j_kt}^M
(I_r₁-A_n-g_r₁n)
-

- d_stG_pp^¢qq^¢^s-s-ss( r₁) (pi_n^¢ | i_kp^¢)P_{i_k-si_ns^¢j_ns^¢j_k-s}^M
(I_r₁-A_n-g_r₁n)
],

(0)
for the two-geminal contributions.

The cross term between the PV^cP and PV^RP equals

áF₀₀^M | PV^cP QR(0)QPV^RP | F₀₀^Mñ =

= -
å
pp^¢ Î R

å
B Î R
Z_B^R
å
ss^¢t
p_s ⁺p_s^¢( 1-| 0_R ñ á 0_R| )×

×
å
kn Î M

å
ii^¢jj^¢ Î {r,l}
((pp^¢ | i_ki_n^¢)-d_ss^¢(pi_n^¢ | i_kp^¢))V_B^{j_nj_k^¢}P_{i_ks^¢i_ns^¢^¢j_ntj_kt^¢}^M.

(0)
In the case when the ZDO approximation is employed for the two-electron integrals we have:

áF₀₀^M | PV^cP QR(0)QPV^RP | F₀₀^Mñ =

= -
å
p Î R

å
B Î R
Z_B^R
å
k Î M

å
i Î { r,l}

å
stt^¢
p_s ⁺p_s ( 1-| 0_R ñ á 0_R| ) ×

×((pp | i_ki_k)-d_ss^¢(pi_k | i_kp))( g_{BA_k}-g_{BB_k}) P_{i_ks^¢i_ks^¢r_ktr_kt}^M.

(0)

The cross term between PV^rrP and PV^RP equals

áF₀₀^M | P V^RPQR(0)QPV^rrP | F₀₀^Mñ =

=
å
k
áF₀₀^M | P V^RPQR(0)QPV^rrP | F₀₀^Mñ_k+

+
å
k ¹ n
áF₀₀^M | PV^RPQR(0)QPV^rrP | F₀₀^Mñ_kn.

(0)
The one-geminal contribution equals

áF₀₀^M | P V^RPQR(0)QPV^rrP | F₀₀^Mñ_k =

= -
å
pp^¢ Î R

å
st
å
\Sbii^¢jj^¢ Î { r,l}

f Î { a,b} \endSb
å
B Î R
Z_B^RV_B^{i_ki_k^¢}v_{pj_k}v_{p^¢j_k^¢}( 1-| 0_R ñ á 0_R| ) ×

×[
å
r₁ Î ImO_R(N_R+1)
p_t | r₁ ñ á r₁| p_t^¢+W_{i_ksi_ks^¢j_ktj_kt^¢}^f_k-t
I_kf-A_r₁-g_{f_kr₁}
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} p_t ⁺| r₁ ñ á r₁| p_t^¢W_{i_ksi_ks^¢[(j)\tilde]_kt [(j)\tilde]_kt^¢}^f_k-t
I_r₁-A_kf-g_{r₁f_k}
],

(0)
while the two-geminal contribution equals

áF₀₀^M | P V^RPQR(0)QPV^rrP | F₀₀^Mñ_kn =

= -
å
pp^¢ Î R

å
s
å
\Sbii^¢jj^¢ Î { r,l}

f Î { a,b} \endSb
å
B Î R
Z_B^RV_B^{i_ki_n^¢}( 1-| 0_R ñ á 0_R| ) ×

×[
å
r₁ Î ImO_R(N_R+1)
p_s | r₁ ñ á r₁|p_s ^¢+W_{i_ksi_ns^¢j_nsj_ks^¢}^f_k-sv_{pj_n}v_{p^¢j_k^¢}
I_kf-A_r₁-g_{f_kr₁}
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢} p_s ⁺| r₁ ñ á r₁| p_s ^¢W_{i_ksi_ns^¢j_ns^¢j_ks}^f_n-sv_{pj_k}v_{p^¢j_n^¢}
I_r₁-A_nf-g_{r₁f_n}
].

(0)
This term, in the case when the ZDO approximation is used for two-electron integrals, equals:

áF₀₀^M | P V^RPQR(0)QPV^rrP | F₀₀^Mñ =

= -
å
pq Î R

å
B Î R
Z_B^R( 1-|0_R ñ á 0_R| )
å
k Î M
å
\Sb jj^¢ Î { r,l}

f Î { a,b} \endSb
å
st
v_{pj_k}v_{qj_k^¢}×

×[
å
r₁ Î ImO_R(N_R+1)

å
s
p_t | r₁ ñ ár₁| q_t ⁺( g_{BA_k}W_{r_ksr_ksj_ktj_kt^¢}^f_k-t+g_{BB_k}W_{l_ksl_ksj_ktj_kt^¢}^f_k-t)
I_kf-A_r₁-g_{f_kr₁}
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢}

å
s
p_t ⁺| r₁ ñ á r₁| q_t ( g_{BA_k}W_{r_ksr_ks [(j)\tilde]_kt[(j)\tilde]_kt^¢}^f_k-t+g_{BB_k}W_{l_ksl_ks[(j)\tilde]_kt[(j)\tilde]_kt^¢}^f_k-t)
I_r₁-A_kf-g_{r₁f_k}
].

(0)
If the energies of electron transfer from (or to) the k-th geminal does not depend on the type of ionized orbital we obtain that the Eq. ( 85) transforms to

áF₀₀^M | P V^RPQR(w)QPV^rrP | F₀₀^Mñ =

= -
å
pq Î R

å
B Î R
Z_B^R( 1-|0_R ñ á 0_R| )
å
k Î M

å
jj^¢ Î { r,l}

å
st
v_{pj_k}v_{qj_k^¢}×

×[
å
r₁ Î ImO_R(N_R+1)

å
s
p_t | r₁ ñ ár₁| q_t ⁺( g_{BA_k}P_{r_ksr_ksj_ktj_kt^¢}^M+g_{BB_k}P_{l_ksl_ksj_ktj_kt^¢}^M)
I_k-A_r₁-g_kr₁
-

-
å
r₁ Î ImO_R(N_R-1)
(-1)^{d_jj^¢}

å
s
p_t ⁺| r₁ ñ á r₁| q_t ( g_{BA_k}P_{r_ksr_ks[(j)\tilde]_kt[(j)\tilde]_kt^¢}^M+g_{BB_k}P_{l_ksl_ks[(j)\tilde]_kt[(j)\tilde]_kt^¢}^M)
I_r₁-A_k-g_r₁k
].

(0)
The last contribution to the effective Hamiltonian of the quantum subsystem is a c-number and can be written as

áF₀₀^M | PV^RP | F₀₀^Mñ = -2
å
A Î M
å
\Sb B Î R B ¹ A\endSbg_ABZ_B^R å
\Sb t_n Î A n Î M\endSb P_n^tt.
(0)
Thereby, we obtained the explicit form of the effective Hamiltonian for the R-subsystem. It allows not only to determine the wave function of the R-subsystem but also to obtain the electronic energy of the whole molecule, which can be written [] as

E₀(q) = E₀^R(q)+E₀₀^M(q),

where E₀^R(q) is the lowest eigenvalue of the effective Hamiltonian Eq. (6) of the R-subsystem and E₀₀^M(q) is the energy of the M-subsystem which is parametrized in the MM form (The detailed transition from the local QM description of molecular electronic structure to the MM will be published elsewhere []).

To demonstrate the importance of transition from the bare Hamiltonian for the R-subsystem to its effective Hamiltonian we have estimated using the above formulae the renormalization of the two-electron parameters of Coulomb interaction (g₁₁ and g₁₂) in the PPP Hamiltonian due to interaction of the p-system with the s-core. In fact the s-p-separation is one of the first examples of separation of electron variables. Incidentally, in this case the contribution of the intersubsystem resonance vanishes for the symmetry reasons. We want to emphasize that the values of correction are independent on the bare values of parameters. We start from the bare value of ~ 7.6 eV for the g₁₂ parameter for the double C-C bond calculated by the formulae accepted in the standard MINDO/3 method and the bare value of ~ 11.8 eV for the g₁₁ parameter which is obtained from atomic spectra[]. We accepted the bond length value 1.339 A which corresponds to that in ethylene. The correction to the g₁₁ and g₁₂ parameters is due to the contribution Eq. (37) to the effective Hamiltonian only. In fact in the ethylene molecule the geminals corresponding to the C-C and C-H bonds contribute to the correction. The numeric estimate of the correction to g₁₁ is negative (the renormalized g₁₁ is smaller than its bare value) and equals to ~ 0.45 eV. This value is not very large which justifies the application of the perturbation theory to correct the bare Hamiltonian. At the same time this value suffies to be important for constructing the Hamiltonian for the p-subsystem. Also we obtained that the rermalization of the g₁₂ parameter is very small (less than 0.3%) and thus can be neglected.

Discussion.

The QM/MM methods become more and more popular in theorerical modeling of large molecular systems. It is caused by rapidly growing needs of biological and organometallic chemistry. The application of the QM/MM schemes to calculations on molecular electronic structure and chemical reactivity is very advantageous. However, the results of up-to-day QM/MM calculations depend on the form of the junction between QM and MM subsystems and, therefore, are ambiguous. All the junction forms proposed in the literature are empirical. It seems to be important to substantiate the form of the junction on the basis of quantum chemical consideration of interaction between the parts of the molecular system treated on the quantum and classical level. In the present work we performed a sequential derivation of the hybrid QM/MM scheme on the basis of a previously developed local description of molecular electronic structure [9]. This allows to express finally the renormalizations of the bare Hamiltonian in terms of the local quantities characterizing the two-electron bonds in the chemically inert i.e. classical subsystem, which can be either calculated or estimated from experimental data on the bond polarizabilities [].

In order to constuct the effective Hamiltonian for the quantum system or equivalently the QM/MM junction we have taken into account both electron transfers between the subsystems and the excitations in the classical part of the molecular system. We have obtained the explicit expressions for this renormalization, which is the function of well-defined characteristics like the polarization propagators of the M-subsystem. Our consideration also required to introduce new quantities X and W which appear due to entanglements of one-electron transfers between the subsystems and excitations leaving the number of electrons in two subsystems unchanged. If the energies of the states with one electron transfers between the two subsystems are replaced with an average value, the quantities X and W in the expressions for the effective Hamiltonian for the quantum subsystem are reduced to the polarization propagators. This allows to express the contributions renormalizing the Coulomb interaction of electrons in the quantum part of the system through such observable quantities as the bond polarizabilities are. The same applies to the corrections renormalizing the one-electron terms of the Hamiltonian for the quantum part. The possibility to express the corrections to the Hamiltonian for the R-subsystem through the experimentally observable quantities such as bond polarizabilities and the ionization potentials of the bonds allows to eliminate the calculations of the electronic structure of the classical part of the whole molecular system and to parameterize them. The above reduction of the QM/MM junction to the sum of transferable contributions from the chemical bonds remaining in the classical part was possible due to the special form for the implicit wave function of electrons underlying the description of the classical part of the system with use of the MM type schemes. We use the APSLG form for this wave function. This provides a route to a possible derivation of the classical (MM) description of the M-subsystem.

With use of the formulae derived above for the effective Hamiltonian for the quantum subsystem we addressed the old problem of substantiation of the parameters of the PPP Hamiltonian (see, for example []). The PPP Hamiltonian for the p-electrons is historically one of the earliest examples of successful separation of electron variables with those in the p-system treated on the QM level, while those in the s-system left to a classical (in fact, MM-like) description. As it is noticed in Freed the The value of parameter g₁₁ in the PPP Hamiltonian equals to 11.06 eV. The value obtained from atomic spectra of carbon is larger on ~ 0.7 eV. This difference can be described by the influence of s-core on the p-subsystem. The correction obtained using written above formulae and MINDO/3 approximation (0.45 eV) is smaller than 0.7 eV but the sign and the order of correction are true. The bare value of the g₁₂ parameter strongly depends on the type of approximation. The correction to this value is small and therefore it is more important to obtain the correct bare value of two-center Coulomb parameter than the correction to it.

References

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[]: I.B. Bersuker, M.K. Leong, J.E. Boggs, and R.S. Pearlman, Int. J. Quantum Chem. 63 (1997) 1051.
[]: CECAM-NSF Meeting on QC/MM methods, Int. J. Quant. Chem 60 (1996) No 6, Special Issue.
[]: T.N. Truong, E.V. Stefanovich, Chem. Phys. Lett. 256 (1996) 348.
[]: A.L. Tchougréeff, Khim. Fiz. 16 (1997) No 6, 62 (in Russian); Chem. Phys. Reps. 16 (1997) 1035 (in English).
[]: A.L. Tchougréeff, Phys. Chem. Chem. Phys. 1 (1999) 1051.
[]: P.-O. Löwdin, J. Math. Phys. 3 (1962) 969; P.-O. Löwdin, in Perturbation Theory and its Applications in Quantum Mechanics, ed. C.H. Wilcox, J. Wiley, NY, 1966.
[]: R.McWeeny, Methods of Molecular Quantum Mechanics, 2nd Edition, AP, London, 1992.
[]: M.B. Darkhovskii, A.L.Tchougréeff, Khim. Fiz. 18 (1999) No 1, 73 (in Russian).
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[]: P.R. Surján, The Concept of the Chemical Bond, in Theoretical Models of Chemical Bonding, Part 2. Ed. Z.B. Maksi\'c, Springer, Heidelberg, 1989, p. 205.
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File translated from T_EX by T_TH, version 2.67.
On 27 Apr 2000, 00:47.

A.M. Tokmachev, A.L. Tchougréeff and I.A. Misurkin Karpov Institute of Physical Chemistry Vorontsovo pole 10, Moscow 103064 RUSSIA

Effective Electronic Hamiltonian for Quantum Subsystem in Hybrid QM/MM Methods as Derived from APSLG Description of Electronic Structure of Classical Part of Molecular System

Abstract

Introduction

Effective Hamiltonian for quantum system

Discussion.

References

A.M. Tokmachev, A.L. Tchougréeff and I.A. Misurkin
Karpov Institute of Physical Chemistry
Vorontsovo pole 10, Moscow 103064 RUSSIA