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Perturbation treatment of multiple site reactivity: Molecule–molecule interactions
Volume 42, number 1
PERTURBATION
CEIEMK!AL PHYSICS LETTERS
TREATMENT
MQLECULE-MOLECULE
OF MULTIPLE
15 August 1976
SITE REACTIVITY:
INTERACTIONS*
Shih-Yung CHANG, tire1 WEINSTEIN
and David CHOU
Deportment of Htarmacology, Mount SinaiScirool of Medicine. City Liitiversity of New York, New York, New York 10029, USA Received 11 April 1976 Revrsed manuscript recerved 19 May 1976 A multrple perturbation method for the treatment of molecular reactivity IS gencrahzed to describe the simultaneous interaction of a molecule contaming several reactive sates with a second molecule and a pomt charge. The modificatron of the reactivity properties of the main molecule by electrostatrc mteraction, polarization, dispersion and correlatron correctlons due to the presence of the second molecule, is identified from the effect of these corrections on the Interaction energy with the point charge The decomposrtron of interaction energies inherent to the perturbation expansron facthtates this analysis. The method is applied to a simple molecule contaming two reactrve sites (formyl fluonde - HCOF), rnteractrng with HF at hydrogen-bond distances. The results are m very good agreement with super-molecule LCAO SCF calcuIatrons with gaussian basis functrons and mdrcate that the redist:rbutron of the electron density caused by the interactron of the two molecules is adequately represented by the perturbatron method.
1. Introduction In a previous publication [I] we formulated the treatment of simultaneous interactions of large molecules with several reactants by a multiple perturbation expansion in which the attacking species were represented by point charges. The method was shown to provide a good characterization of diprotonation in adenine and to account successfully for polarizatmn corrections to the electrostatic interaction of the point charges with the molecule. The effects of the interaction with a neutral molecule on the reactivity of adenine were also reproduced successfully by this perturbation method [2] _ To eliminate the need for a point charge expansion and in order to allow for the relaxatian of the charge distribution in both interacting molecules, we present here a generalization of the multiple perturbation approach to the case in which the main molecule (A) that contains several reactive sites is attacked by a second molecule (B) and by a * This work was supported in part by Grant MN-17489 from the National instrtute of Mental Health.
point charge perturbation (e-g_, a proton) that serves to probe the reactr-vity characteristics of A. The theoretical treatment seeks a simple description cf the effects that an interaction with B will have on the reactivity of the main molecule A toward a second attacking species, represented by a perturbation W. As detailed in ref. [ l] , the practical purpose of the method is to analyze the reactivity of very large molecu!es interacting at drstances at which the approximation of “small orbital overlap” [3] is valid. Because it treats each molecule individually, the perturbation method can be applied to large molecular complexes that are impractical for “super-molecule” calculations. It should be most useful for the identification of molecular structural factors affecting reactivity because the interaction is expressed directly in terms of separate contributions from perturbed molecules. For cases in which accuracies beyond SCF are necessary, this multiple perturbation approach also provides terms that account for certain specific contributions such as dispersion and intramolecular correlation. An application of the generalized multrple perturbation method is illustrated with fortnyl fluoride 145
Volume42, number1
CHEhfICAL PHYSICS LETTERS
(HCOF) as an exampIe of a molecule A that contains two different nucleophilic sites, HF as the attacking molecule B and a point charge perturbation W. The resdts are compared to ab initio LCAO SCF calculations of the corresponding super-molecules composed of A and B.
15 August 1976
The total wavefunction for the system satisfies &19 = tZ\k,and can be expanded in a double perturbation series (for a review see ref. (41) with
(10) and
2. Theory The system composed of a molecule A containing several reactive sites and a second attacking molecule (B), is represented by the hamdtonian ffAB =HA
+I$
f
qnt,
(1)
where HA and HB are the molecular harniltonians of the separate molecules. In terms of the independent particle SCF hamiltonians of the isolated molecules (I’ and I$), the hamiltonian HAB can be written as HAB=IPA+~+(~/A+++H,S=~B+YI/AB,
(2)
where @&
= E:&
(3)
and &&:B
= E$k&,
(41
with @& defined as the roduct of the antisymmetrized 8 wavefunctions & and Qg &B = &&
(51
Then EiB=E$-Eg
(6)
and the SW energies of the separate molecules are @*Cc’ = Q&LHA1#~k
EsSCF = <&I&I&.
(11) where the pnmes in eqs. (10) and (11) indicate that terms with m= n = 0 are not included. The simple product form used in (5) is also assumed for (Xk)AB of eq. (LO), which becomes n
m
(12) It should be noted that the use of simple product forms for &B and (XkIA* is not an approximation of the expansion in (10) and (11) but the total wavefunctlon for the system remains a sum of products of separately antisymmetrized wavefunctions. Substituting (10) and (11) into the Schrodinger equation and collecting same order terms one obtains a series of perturbation equations that are solved by introducing two orthonormal bases composed of SCF molecular orbitals of A and B, respectively. The perturbation wavefunctions (X2)* and (Xg)B in eq. (12) are then expanded into determinants formed by excitations from SCF configurations of A and B. For convenience the excitations are restricted to A + A and B + B types only. The following energy components are obtained in terms of the spin orbitak:
(7)
Addmg the external perturbation W by a point charge Q to (2), one obtains n=HO,
+pv*J.J +hW,
where W is of the form
occ unocc
(8)
(Et)AB
= E& + E$B =
C C
isA
PEA
(Ei -
$)-l
(PIWIl12
oaz llrloaz
(14)
146
Volume 42, number 1
I.5 August 1976
CHEMICAL PEWSKS LETTERS
x (Ibvlp~(jltFlq~~pqllij), occ unocc
Ei(m)~4~f=~
pgA
occ iz
w-w unocc ,cB
(ei-ep)-’
(cj-Eq)-’ Wb)
XOiwl~>(p4lij~. The definition
of the operators
in eqs. (13)-(18)
is:
with nuclear charges Z, and Za; Jk is a one-electron coulombic operator and ei represents the energy of molecular otbltal i. The matrix element (pqllij) is defined as
(20) -
s
p*(i)j(l)rii
qt(2)i(2)dr1
drz.
The physical interpretation of the perturbation energies in eqs. (13)-(18) arallels that of the cotreP spending terms @~,I?$, El, .... etc.) in ref. [l]. ERA and #!?& in eq. (13) are simply the electrostatic mteraction energies between the point charge perturbation W and the unperturbed molecules A and B. E$A and @B in eq. (14) are the contributions of the isolated molecules A and B to the polarization energy from the attack of W; these terms correspond to E~,o in ref. 11). EVA and E:B in eq. (17) are the corrections to the electrostatic interaction energies between the point charge perturbation W and the two interacting molecules A and B. This correction accounts, to first order, for the moditication induced in the electrostatic potential felt by the point charge approaching each of the two molecules [5,6] by the presence of the other molecule. It corresponds to the E1,1 term in ref. [I] and is the first order cotrection due to the simultaneous attack of A by molecule B and the perturbation W. The first and second order terms (Ek)AB and (Ei)AB in eqs. (15) and (15) represent the pertutbation expansions of the interaction energy of molecules A and B. (Ek)AB contains the correction e$ due to 147
Volume 42, nun&x
CHEMICAL PHYSICS LETTERS
1
the purely electrostatic interaction unperturbed molecules, such that
between the two
(21) where HAB, & and EisCF are defined in eqs. (2)-(7). The second order interaction term (Eg)Au in eq. (16) contmns contributions
* Ei + Ei i- (E&
+ Ek(le)
+ .&n(le),
(22)
where (le) signifies the one-electron part of the terms in eq. (16a). For comparison with an SCF calculation of the super-molecule, the interaction of Athe A.33 system with the perturbation FVwill be approximated by
(23) where EiA(le) and E&(Ie) represent the one-electron part of E.$* and E& rn eq. (1 Sa), and include only terms involving A-B interaction corrections of the form uA + xgg~ + The additional terms in eqs. (16) and (181, such as the two-electron terms involving the intramolecular <~jl$ and intermolecular (~4153~ represent contributions to corrections beyond SCF’_ The approximation introduced in the interaction * For a simlhr dccomposrtron of the SCF energy in the supermolecule see refs. (7,s 1.
l&8
energy between A and B by the separate antisyrnrnetrization of the wavefunctions for molecule A and molecule B, remains uncorrected. This makes the method efficiently applicable to large systems but, in principle, reduces its accuracy compared to the method of Murrell and co-workers 131.
from three different types of
adjustments in molecules A and B: (i) the polarization response of each molecule to the presence of the other, represented by one-electron integrals involving uA + Xz& Jk; (ii) the electron correlation within each of the molecules, represented by integrals of the type (&i/ 1 in -4 and B, separately; and (iii) the dispersion interaction between A and B, represented by the term E&B) in eq. (16). The mixed term (Ei)As in eq. (18) represents the first order correction to the polarization response of the interacting A3 system to the perturbation by W. The physical interpretation of the contributrons in this correction is obtained from points (r)-(iii) in the description of (Eg)AB . Thus, the mutual polarization of A and B, the intramolecular correlation and the “correlated” dispersion behavior of electrons in A and B, all contribute to correct the polarization response of the AB system to the perturbation IV. The super-molecule SCF calculatron of the AB system may be approximated by E;%
15 August i976
3. Application of the method to interactions formyl fluoride
with
The calculations were performed ab-initio with the POLYATOM’ LCAO SCF MO program, using the (Ss, 3p, 2~~) gaussian basis Set of Whitman and Homback [9] contracted to minimal basis. The molecules remained fixed in the initial geometries. The geometry for formyl fluoride (HCOF) is from ref. [lo] and HF was calculated with a bond length of 1.733 au. The interaction of HF with HCOF was calculated at three positions, as shown in fig. 1. These interaction geometries correspond to the hydrogen bonding configurations in (HF), [ 1 l] and (H2 C) _.HF) [12], with 81 =d2 = 4’; R 1 = R2 = 5 au and a colinear 3 ... HF bond with R3 = 5.159 au. Table i presents the terms in the perturbation calculation of the interaction between HCOF and HF at each of the three positions indicated in fig. 1. The values of the total energy calculated for the complex (HCOF...HF) from eq. (22) are compared to full supermolecule SCF calculations. The agreement between the two types of results is surprisingly good and indicates that in this system the combined neglect of “exchange” and “charge transfer” terms is a very good approximation at hydrogen-bonding distances. The interactions with a positive point charge (W) were calculated at points of minimum in the electrostatic potential [5,6] generated by the unperturbed molecule A (i.e. HCOF). The position of these points is indicated in fig. l_ Some of the results obtained for the interaction wrth the positive point charge are given in tables 2 and 3. The first order interaction energy between the isolated molecules and the perturbation W (commonly represented as the “electrostatic potential” [5,6]), is * The POLYATOM program package was obtained from the Quantum Chemistry Program Exchvlge (QCPE 1=238), Indiana University.
Volume 42, number 1
15 August 1976
CHElCiICAL PHYSICS LETTERS
Table 1 Perturbation expansion of energies in the HCOF...HF compkx
F
/“ /
**
posltKWl 3
Ii
a mm. b
Energy termb)
Interaction geometrya) 1
min. a
E&F
-279.033830
-279.033830
-279.033830
- 104.821455
- I 04.82 1485
- 104.821485
EL
-
38.236115
-
40.078017
-
36.283879
E&J E&3)
-
37.500681 37 537618
-
39.238310 39.362604
-
35.455906
E&
-
0.000615
-
0.000600
-
E&(le)
-
0.000002
-
0.000003
-
il
110.533905
35 661361 0.001025 0.000092 108.408092
112.288482
ES
-311.521205
-311.521159
-311.526764
Em SCF
-311.521359
-311.521587
-311.526436
&$$p
iti
3
E&l-
Em nud&r \
2
-
6.1
-
6.2
-
9.3
-
6.0
-
6.0
-
95
4 See fig- I for definition of geometries. b) Energies given in harnees. c, aE is the complex stabthzatron enemy. in kcal/moIe. SCF energies for the separate molecuks are EgsF = -211.883478 hdrtree; Es = -99.628178 hartree.
posItloll 1
F
Fig. 1. The three different positions of HF mteracting with HCOF. The positions of the electrostatic potential minima (I and b at which the perturbation IV is placed, is indicated by 0.
given in column A + B of Ey. This term is corrected the E: contributions to account for the interaction
A and B. In all three positions examined, we found the correction introduced by Ei to be in the direction needed to bring the result closer to the interaction energy EF(SCF) obtained from the super-molecule SCF calculation. The super-molecule term is given between
by
Tabb 2 Complex m geometry 2: expansion of interactton with IV Energy termb)
Posltion of rd
Molecule
Eq
a
E:+E:
b a b
-0.036807 -0.078062 -0.044972 -0.072687
E2 0
a
-0 027885
-0.06850+
b
-0.067380
a b
-0 026275 -0.067576
E$+E;
A
A+B
Super-molecule (AB)
0.412581 0.009043
0.375774 -0.069019
Ef
a 0.369759 b -0.062568
0.413433 0.009089
E:
-0.000182
0.368461 -0.063599 -0.C96389 -0.067562
a -0.090931 b -0.067643
-0.068509 -0.000182
-0.094784 -0.067758
B
a) See fg. 1 for the definition of W posttrons. b) Energies given in hartrees.
149
VoIurw
42, numixx 1
Table 3 Complex
in geometry
Energy
term
3: expansion POSltiO~ of w
of interaction Molecule A
B
-0 036807
0.009692
--0.078062
a.175909
-0.032462 -0.094569
0.009962 0.179355
A+B
Super-molecule (AB)
-0.027115
E; II -0.021153
0.097847
-0.027885
-0.000198
-0.028083
-0.032279 -0.000202
-0.099659 -0.028412
-0.065247
-0.031649
-0.096896
the calculation of the polarization term Et. The corrections diminish the absolute value of &(A + B) in a of tabIe 2 and b of table 3. The absoiute value is properIy corrected upwards by the correction in b of table 2 and in (I of table 3. In all the cases studied, the corrected value [Ei +E$(le)] was closer to +Jle@(SCF). The agreement obtained between the polarization terms calculated by the perturbation expansion and
the corresponding terms in the super-moIecule caiculation is very good. lks serves as an indication that the of the electron density in the HCOF .._ HF complex due to the interaction between the two fragments is adequately represented by the perturbation method. The reactivity characteristics of the interacting molecules, that are obtainable from this approach, should therefore be reliable and very useful in the analysis of multiple site reactivity_ The correlation and dispersion corrections to these reactivity characteristics are studied now with the use of higher order terms in the perturbstlon expansion [13] _ An extension of this perturbation expansion for the treatment of a large number of simultaneously interacting molecules is easily obtainable along the lines presented here; an application to salvation studies of large molecules ~lti be presented in a forthcoming report [13].
b
0.091585
-0.022500 0.084786
-0.067380 -0 028210
mokcule, the Ei correcticin diminishes it (e.g., a and b of table 2 and II and B of table 3). The same correct effect of the higher order correction is also evident in
150
August1976
with W
in the last column of tables 2 and 3. The tables show that whenever the absoIute value of the Ef term in the (A + B) column is larger than @(SCF) of the super-
redistribution
15
CHEMICALPHYSICSLETTERS
E2 a -0.028577 b -0.093254
Acknowledgement We thank Dr. J-E. Eilers for very helpful discussions. The computations were supported in part by a grant of computer time from the Central Computer
Facility of the City University
of New York.
References
111R.J. Bartlett and H. Weinstein, Chem. Phys. Letters 30 (1975) 441.
VI H. Weinstem, Intern. 5. Quantum Biol. QBS2 (1975) 59, and 131 J.N. Sot. I41 J.O.
151
161 [71 ISI
191 [lOl 1111 [121 (131
references therein. MurrelI, M. Randlc and D R. Williams, Proc. Roy. A284 (1965) 331. Hnschfelder, W. Byers Brown and S. Epstein, in:
Advances in quantum chemistry, Vol. 1. ed. P.-O. L0wdm (Academic Press, New York, 1964) p. 255. E. Scrocco and J. Tom&, TOPICSm current chemistry, new concepts II, No. 42 (Springer, Berlin, 1973) p. 95, and references therem S. Srebremk, H. Wemstein and R. Pauncz, Chem. Phys. Letters 20 (1973) 419. K. Morokuma, J. Chem. Phys. 55 (1971) 1236. I;. Morokuma, S. Iwata and W-A. Lathan, in: The world of quantum chemistry, eds. R. Daudel and 8. Pullman (Reidel, Dordrecht. 1974) p- 277. D-R. Whitman and C J. Homback, J. Chem. Phys 5 1 (1969) 398. P. Favero, A.M. Mirn and J.G. Beker, J. Chem. Phys. 31 (1959) 566. D-R. Yarkoni, S.V. O’Neil, H.F. Schaefer III, C-P. Baskin and C.F. Bender, J. Chem. Phys. 60 (1974) 85.5. P. Kollman, J. McKelvey, A. Johansson and S. Rothenberg. I_ Am. Chem. Sot. 97 (1975) 955. S-Y. Chang, H. Weinstein and D. Chou, to be published.
PERTURBATION
CEIEMK!AL PHYSICS LETTERS
TREATMENT
MQLECULE-MOLECULE
OF MULTIPLE
15 August 1976
SITE REACTIVITY:
INTERACTIONS*
Shih-Yung CHANG, tire1 WEINSTEIN
and David CHOU
Deportment of Htarmacology, Mount SinaiScirool of Medicine. City Liitiversity of New York, New York, New York 10029, USA Received 11 April 1976 Revrsed manuscript recerved 19 May 1976 A multrple perturbation method for the treatment of molecular reactivity IS gencrahzed to describe the simultaneous interaction of a molecule contaming several reactive sates with a second molecule and a pomt charge. The modificatron of the reactivity properties of the main molecule by electrostatrc mteraction, polarization, dispersion and correlatron correctlons due to the presence of the second molecule, is identified from the effect of these corrections on the Interaction energy with the point charge The decomposrtron of interaction energies inherent to the perturbation expansron facthtates this analysis. The method is applied to a simple molecule contaming two reactrve sites (formyl fluonde - HCOF), rnteractrng with HF at hydrogen-bond distances. The results are m very good agreement with super-molecule LCAO SCF calcuIatrons with gaussian basis functrons and mdrcate that the redist:rbutron of the electron density caused by the interactron of the two molecules is adequately represented by the perturbatron method.
1. Introduction In a previous publication [I] we formulated the treatment of simultaneous interactions of large molecules with several reactants by a multiple perturbation expansion in which the attacking species were represented by point charges. The method was shown to provide a good characterization of diprotonation in adenine and to account successfully for polarizatmn corrections to the electrostatic interaction of the point charges with the molecule. The effects of the interaction with a neutral molecule on the reactivity of adenine were also reproduced successfully by this perturbation method [2] _ To eliminate the need for a point charge expansion and in order to allow for the relaxatian of the charge distribution in both interacting molecules, we present here a generalization of the multiple perturbation approach to the case in which the main molecule (A) that contains several reactive sites is attacked by a second molecule (B) and by a * This work was supported in part by Grant MN-17489 from the National instrtute of Mental Health.
point charge perturbation (e-g_, a proton) that serves to probe the reactr-vity characteristics of A. The theoretical treatment seeks a simple description cf the effects that an interaction with B will have on the reactivity of the main molecule A toward a second attacking species, represented by a perturbation W. As detailed in ref. [ l] , the practical purpose of the method is to analyze the reactivity of very large molecu!es interacting at drstances at which the approximation of “small orbital overlap” [3] is valid. Because it treats each molecule individually, the perturbation method can be applied to large molecular complexes that are impractical for “super-molecule” calculations. It should be most useful for the identification of molecular structural factors affecting reactivity because the interaction is expressed directly in terms of separate contributions from perturbed molecules. For cases in which accuracies beyond SCF are necessary, this multiple perturbation approach also provides terms that account for certain specific contributions such as dispersion and intramolecular correlation. An application of the generalized multrple perturbation method is illustrated with fortnyl fluoride 145
Volume42, number1
CHEhfICAL PHYSICS LETTERS
(HCOF) as an exampIe of a molecule A that contains two different nucleophilic sites, HF as the attacking molecule B and a point charge perturbation W. The resdts are compared to ab initio LCAO SCF calculations of the corresponding super-molecules composed of A and B.
15 August 1976
The total wavefunction for the system satisfies &19 = tZ\k,and can be expanded in a double perturbation series (for a review see ref. (41) with
(10) and
2. Theory The system composed of a molecule A containing several reactive sites and a second attacking molecule (B), is represented by the hamdtonian ffAB =HA
+I$
f
qnt,
(1)
where HA and HB are the molecular harniltonians of the separate molecules. In terms of the independent particle SCF hamiltonians of the isolated molecules (I’ and I$), the hamiltonian HAB can be written as HAB=IPA+~+(~/A+++H,S=~B+YI/AB,
(2)
where @&
= E:&
(3)
and &&:B
= E$k&,
(41
with @& defined as the roduct of the antisymmetrized 8 wavefunctions & and Qg &B = &&
(51
Then EiB=E$-Eg
(6)
and the SW energies of the separate molecules are @*Cc’ = Q&LHA1#~k
EsSCF = <&I&I&.
(11) where the pnmes in eqs. (10) and (11) indicate that terms with m= n = 0 are not included. The simple product form used in (5) is also assumed for (Xk)AB of eq. (LO), which becomes n
m
(12) It should be noted that the use of simple product forms for &B and (XkIA* is not an approximation of the expansion in (10) and (11) but the total wavefunctlon for the system remains a sum of products of separately antisymmetrized wavefunctions. Substituting (10) and (11) into the Schrodinger equation and collecting same order terms one obtains a series of perturbation equations that are solved by introducing two orthonormal bases composed of SCF molecular orbitals of A and B, respectively. The perturbation wavefunctions (X2)* and (Xg)B in eq. (12) are then expanded into determinants formed by excitations from SCF configurations of A and B. For convenience the excitations are restricted to A + A and B + B types only. The following energy components are obtained in terms of the spin orbitak:
(7)
Addmg the external perturbation W by a point charge Q to (2), one obtains n=HO,
+pv*J.J +hW,
where W is of the form
occ unocc
(8)
(Et)AB
= E& + E$B =
C C
isA
PEA
(Ei -
$)-l
(PIWIl12
oaz llrloaz
(14)
146
Volume 42, number 1
I.5 August 1976
CHEMICAL PEWSKS LETTERS
x (Ibvlp~(jltFlq~~pqllij), occ unocc
Ei(m)~4~f=~
pgA
occ iz
w-w unocc ,cB
(ei-ep)-’
(cj-Eq)-’ Wb)
X
of the operators
in eqs. (13)-(18)
is:
with nuclear charges Z, and Za; Jk is a one-electron coulombic operator and ei represents the energy of molecular otbltal i. The matrix element (pqllij) is defined as
(20) -
s
p*(i)j(l)rii
qt(2)i(2)dr1
drz.
The physical interpretation of the perturbation energies in eqs. (13)-(18) arallels that of the cotreP spending terms @~,I?$, El, .... etc.) in ref. [l]. ERA and #!?& in eq. (13) are simply the electrostatic mteraction energies between the point charge perturbation W and the unperturbed molecules A and B. E$A and @B in eq. (14) are the contributions of the isolated molecules A and B to the polarization energy from the attack of W; these terms correspond to E~,o in ref. 11). EVA and E:B in eq. (17) are the corrections to the electrostatic interaction energies between the point charge perturbation W and the two interacting molecules A and B. This correction accounts, to first order, for the moditication induced in the electrostatic potential felt by the point charge approaching each of the two molecules [5,6] by the presence of the other molecule. It corresponds to the E1,1 term in ref. [I] and is the first order cotrection due to the simultaneous attack of A by molecule B and the perturbation W. The first and second order terms (Ek)AB and (Ei)AB in eqs. (15) and (15) represent the pertutbation expansions of the interaction energy of molecules A and B. (Ek)AB contains the correction e$ due to 147
Volume 42, nun&x
CHEMICAL PHYSICS LETTERS
1
the purely electrostatic interaction unperturbed molecules, such that
between the two
(21) where HAB, & and EisCF are defined in eqs. (2)-(7). The second order interaction term (Eg)Au in eq. (16) contmns contributions
* Ei + Ei i- (E&
+ Ek(le)
+ .&n(le),
(22)
where (le) signifies the one-electron part of the terms in eq. (16a). For comparison with an SCF calculation of the super-molecule, the interaction of Athe A.33 system with the perturbation FVwill be approximated by
(23) where EiA(le) and E&(Ie) represent the one-electron part of E.$* and E& rn eq. (1 Sa), and include only terms involving A-B interaction corrections of the form uA + xgg~ + The additional terms in eqs. (16) and (181, such as the two-electron terms involving the intramolecular <~jl$ and intermolecular (~4153~ represent contributions to corrections beyond SCF’_ The approximation introduced in the interaction * For a simlhr dccomposrtron of the SCF energy in the supermolecule see refs. (7,s 1.
l&8
energy between A and B by the separate antisyrnrnetrization of the wavefunctions for molecule A and molecule B, remains uncorrected. This makes the method efficiently applicable to large systems but, in principle, reduces its accuracy compared to the method of Murrell and co-workers 131.
from three different types of
adjustments in molecules A and B: (i) the polarization response of each molecule to the presence of the other, represented by one-electron integrals involving uA + Xz& Jk; (ii) the electron correlation within each of the molecules, represented by integrals of the type (&i/ 1 in -4 and B, separately; and (iii) the dispersion interaction between A and B, represented by the term E&B) in eq. (16). The mixed term (Ei)As in eq. (18) represents the first order correction to the polarization response of the interacting A3 system to the perturbation by W. The physical interpretation of the contributrons in this correction is obtained from points (r)-(iii) in the description of (Eg)AB . Thus, the mutual polarization of A and B, the intramolecular correlation and the “correlated” dispersion behavior of electrons in A and B, all contribute to correct the polarization response of the AB system to the perturbation IV. The super-molecule SCF calculatron of the AB system may be approximated by E;%
15 August i976
3. Application of the method to interactions formyl fluoride
with
The calculations were performed ab-initio with the POLYATOM’ LCAO SCF MO program, using the (Ss, 3p, 2~~) gaussian basis Set of Whitman and Homback [9] contracted to minimal basis. The molecules remained fixed in the initial geometries. The geometry for formyl fluoride (HCOF) is from ref. [lo] and HF was calculated with a bond length of 1.733 au. The interaction of HF with HCOF was calculated at three positions, as shown in fig. 1. These interaction geometries correspond to the hydrogen bonding configurations in (HF), [ 1 l] and (H2 C) _.HF) [12], with 81 =d2 = 4’; R 1 = R2 = 5 au and a colinear 3 ... HF bond with R3 = 5.159 au. Table i presents the terms in the perturbation calculation of the interaction between HCOF and HF at each of the three positions indicated in fig. 1. The values of the total energy calculated for the complex (HCOF...HF) from eq. (22) are compared to full supermolecule SCF calculations. The agreement between the two types of results is surprisingly good and indicates that in this system the combined neglect of “exchange” and “charge transfer” terms is a very good approximation at hydrogen-bonding distances. The interactions with a positive point charge (W) were calculated at points of minimum in the electrostatic potential [5,6] generated by the unperturbed molecule A (i.e. HCOF). The position of these points is indicated in fig. l_ Some of the results obtained for the interaction wrth the positive point charge are given in tables 2 and 3. The first order interaction energy between the isolated molecules and the perturbation W (commonly represented as the “electrostatic potential” [5,6]), is * The POLYATOM program package was obtained from the Quantum Chemistry Program Exchvlge (QCPE 1=238), Indiana University.
Volume 42, number 1
15 August 1976
CHElCiICAL PHYSICS LETTERS
Table 1 Perturbation expansion of energies in the HCOF...HF compkx
F
/“ /
**
posltKWl 3
Ii
a mm. b
Energy termb)
Interaction geometrya) 1
min. a
E&F
-279.033830
-279.033830
-279.033830
- 104.821455
- I 04.82 1485
- 104.821485
EL
-
38.236115
-
40.078017
-
36.283879
E&J E&3)
-
37.500681 37 537618
-
39.238310 39.362604
-
35.455906
E&
-
0.000615
-
0.000600
-
E&(le)
-
0.000002
-
0.000003
-
il
110.533905
35 661361 0.001025 0.000092 108.408092
112.288482
ES
-311.521205
-311.521159
-311.526764
Em SCF
-311.521359
-311.521587
-311.526436
&$$p
iti
3
E&l-
Em nud&r \
2
-
6.1
-
6.2
-
9.3
-
6.0
-
6.0
-
95
4 See fig- I for definition of geometries. b) Energies given in harnees. c, aE is the complex stabthzatron enemy. in kcal/moIe. SCF energies for the separate molecuks are EgsF = -211.883478 hdrtree; Es = -99.628178 hartree.
posItloll 1
F
Fig. 1. The three different positions of HF mteracting with HCOF. The positions of the electrostatic potential minima (I and b at which the perturbation IV is placed, is indicated by 0.
given in column A + B of Ey. This term is corrected the E: contributions to account for the interaction
A and B. In all three positions examined, we found the correction introduced by Ei to be in the direction needed to bring the result closer to the interaction energy EF(SCF) obtained from the super-molecule SCF calculation. The super-molecule term is given between
by
Tabb 2 Complex m geometry 2: expansion of interactton with IV Energy termb)
Posltion of rd
Molecule
Eq
a
E:+E:
b a b
-0.036807 -0.078062 -0.044972 -0.072687
E2 0
a
-0 027885
-0.06850+
b
-0.067380
a b
-0 026275 -0.067576
E$+E;
A
A+B
Super-molecule (AB)
0.412581 0.009043
0.375774 -0.069019
Ef
a 0.369759 b -0.062568
0.413433 0.009089
E:
-0.000182
0.368461 -0.063599 -0.C96389 -0.067562
a -0.090931 b -0.067643
-0.068509 -0.000182
-0.094784 -0.067758
B
a) See fg. 1 for the definition of W posttrons. b) Energies given in hartrees.
149
VoIurw
42, numixx 1
Table 3 Complex
in geometry
Energy
term
3: expansion POSltiO~ of w
of interaction Molecule A
B
-0 036807
0.009692
--0.078062
a.175909
-0.032462 -0.094569
0.009962 0.179355
A+B
Super-molecule (AB)
-0.027115
E; II -0.021153
0.097847
-0.027885
-0.000198
-0.028083
-0.032279 -0.000202
-0.099659 -0.028412
-0.065247
-0.031649
-0.096896
the calculation of the polarization term Et. The corrections diminish the absolute value of &(A + B) in a of tabIe 2 and b of table 3. The absoiute value is properIy corrected upwards by the correction in b of table 2 and in (I of table 3. In all the cases studied, the corrected value [Ei +E$(le)] was closer to +Jle@(SCF). The agreement obtained between the polarization terms calculated by the perturbation expansion and
the corresponding terms in the super-moIecule caiculation is very good. lks serves as an indication that the of the electron density in the HCOF .._ HF complex due to the interaction between the two fragments is adequately represented by the perturbation method. The reactivity characteristics of the interacting molecules, that are obtainable from this approach, should therefore be reliable and very useful in the analysis of multiple site reactivity_ The correlation and dispersion corrections to these reactivity characteristics are studied now with the use of higher order terms in the perturbstlon expansion [13] _ An extension of this perturbation expansion for the treatment of a large number of simultaneously interacting molecules is easily obtainable along the lines presented here; an application to salvation studies of large molecules ~lti be presented in a forthcoming report [13].
b
0.091585
-0.022500 0.084786
-0.067380 -0 028210
mokcule, the Ei correcticin diminishes it (e.g., a and b of table 2 and II and B of table 3). The same correct effect of the higher order correction is also evident in
150
August1976
with W
in the last column of tables 2 and 3. The tables show that whenever the absoIute value of the Ef term in the (A + B) column is larger than @(SCF) of the super-
redistribution
15
CHEMICALPHYSICSLETTERS
E2 a -0.028577 b -0.093254
Acknowledgement We thank Dr. J-E. Eilers for very helpful discussions. The computations were supported in part by a grant of computer time from the Central Computer
Facility of the City University
of New York.
References
111R.J. Bartlett and H. Weinstein, Chem. Phys. Letters 30 (1975) 441.
VI H. Weinstem, Intern. 5. Quantum Biol. QBS2 (1975) 59, and 131 J.N. Sot. I41 J.O.
151
161 [71 ISI
191 [lOl 1111 [121 (131
references therein. MurrelI, M. Randlc and D R. Williams, Proc. Roy. A284 (1965) 331. Hnschfelder, W. Byers Brown and S. Epstein, in:
Advances in quantum chemistry, Vol. 1. ed. P.-O. L0wdm (Academic Press, New York, 1964) p. 255. E. Scrocco and J. Tom&, TOPICSm current chemistry, new concepts II, No. 42 (Springer, Berlin, 1973) p. 95, and references therem S. Srebremk, H. Wemstein and R. Pauncz, Chem. Phys. Letters 20 (1973) 419. K. Morokuma, J. Chem. Phys. 55 (1971) 1236. I;. Morokuma, S. Iwata and W-A. Lathan, in: The world of quantum chemistry, eds. R. Daudel and 8. Pullman (Reidel, Dordrecht. 1974) p- 277. D-R. Whitman and C J. Homback, J. Chem. Phys 5 1 (1969) 398. P. Favero, A.M. Mirn and J.G. Beker, J. Chem. Phys. 31 (1959) 566. D-R. Yarkoni, S.V. O’Neil, H.F. Schaefer III, C-P. Baskin and C.F. Bender, J. Chem. Phys. 60 (1974) 85.5. P. Kollman, J. McKelvey, A. Johansson and S. Rothenberg. I_ Am. Chem. Sot. 97 (1975) 955. S-Y. Chang, H. Weinstein and D. Chou, to be published.