- Email: [email protected]
Second-order properties of bonds
Volume 22; number 3
15 October
CHEMICAL PHYSICS LETTERS
1973
SECOND-ORDERPROPERTIESOFBONDS A.T. AMOS and R.J. CRISPIN Depurtnrent
ojhlathemotics, The Unilerrity, Notringlram, UK
Received 26 July 1973
By defining a localized hamiltonian for a saturated molecule its second order molecular properties may be regarded as a sum of bond contributions. Static and dj-aamic pokuizabilities are calculated for the C-H bond in methane and the latter one used to calculate the van der Waals interaction between two CH4 molecules.
1. Introduction
For many years, and even before the advent of Mechanics, it has been known empirically that &properties of certain classes of molecules can be found by adding together contributions from the bonds and atoms in these molecules. Thus, to give an Quantum
example, the po!arizabiIity of a saturated hydrocarbon can be found by adding together empirical polarizabilities of the individual CC and CH bonds [l] I Naturally, as Quantum Mechanics began to be applied to chemical properties, there were many attempts to give a theoretical foundation to this empirical resuIt. Some of these attempts have been reasonably successful but none of them completely so, pstly because it is only recently that sufficiently accurate molecular wavefunctions have become available and partly because the. application of perturbation theory within the context of the Hartree-Fock scheme has
been fully understood only in the past few years. Now that these two difficulties have been removed it seems time for a reconsideration of the problem. Indeed, to some extent this has already begun with the fine work by Ruedenberg, England and their collaborators (for some references see ref. [2]). These, however, have in the main considered first-order properties, e.g., dip& moments [3] _Although there have been interesting discussions of second-order properties, e.g., polarizabilities and susceptibilities, the methods used have not been suitable for computing numerical values 530
of the second-order bond properties [4,5]. In this note we wish to suggest a method which can easily be used to define and to calculate values of the bond properties of CH and CC bonds in saturated hydrocarbons where reasonably accurate HartreeFock wavefunctions are available. As examples we use the method to fiid the static polarizabilities of the CH bond in methane and the van der Waals interaction
betyeen hvo methane-like CH bonds.
2. Bond orbitals Suppose we consider a molecule with localized bonds, e.g., a saturated hydrocarbon, and no delocalized electrons. L-eta0 be the ground-state Hartree-Fock wavefunction built up ofn doubly occupied molecular orbitals {cji), i= 1, ..., II, i.e.,
9’0 = I @la &P ..‘ $,a &jPI I
(1)
where the 4; satisfy hO@i= ei@i, 110being
t52
Normally
(2) Hartree-Fock the
hamiltonian.
{I#+-: are not themselves
localized
or-
bitals but they can bi transfarmed to a new and equivalent set {we), 01= 1, m-b,12,whic+re localized along the bonds and within the inner shells of atoms [6]. This transformation is the basis of .d theories of
Volume 22, number 3
bond properties *. Following the suggestion of Amos and Musher [V] which is analogous to the method used by Diner et al. [lo] we define a new one-electron operator 11; by n
C
hb =ho-
15 October
CHEMICAL PHYSICS LETTERS
where the changes in the orbit& are obtained from the equations (/zb - E;) NJ;.= (EJ - 2) Wj .
e4111wp>(waI
Pf”
in the Dirac notation, with ecro = (~,lh~l~~). hb and hr-,have the sameVirtual orbitals but the lowest JZ orbit& of Izb are the localized {w,) whereas the II lowest orbitals of ho are the canonical ones {gi]s Using the izb we can now define
1973
the k-electron
operator Hb by 2n (4)
and this has the property that
(8)
This equation can be solved by the sum-over-states method [ 111 if, in the original calculations ofqg, enou& basis orbit& are used to enable the virtual orbitals of 11, to form a reasonably complete set for the problem. As ;1 rule this will not be the case, and then variational-perturbation t,echniques [I l] can be used. Provided each rvi is chosen to be orthogonal to the unperturbed orbital rui the second-order change in the energy due to the perturbntion is @en by n E, =2~
{~Vi(Z
(9)
IWI_).
(5) Using (9) we can define the contribution from the ith bond orbital as
where we now write ‘k, in the transformed form XIJo= IWlcz
rvrp ... W,,LyW,pI
(6) E; = 2 (W&ZI\+,
and E, = 2 5
(10)
with the property that the total second-order energy will be the sum of these bond contributions ‘, i.e., Ei .
i=l
3. Perturbation
theory
Normally the second-order property is defied
Now we apply uncoupled perturbation theory. Let the perturbation be represented by the sum of one. electron operators, e.g., 2n
LyI.=-4(Wi(Z(WI).
w-gz(i).
(7)
The first-order wavefunction is given by
* There are many ical orbit& to
as -2E,, e.g., the components of the po!arizabWy or susceptibility tensor. If crj is the contribution from the ith bond-orbital to the second-order property, then
schemes
for transforminS
from
02)
Clearly there are correcting terms to be added to this expression. These arise essentially because ‘IX0is the Hartree-Fock wavefunction and not the correct onewith the consequence that HA is not the proper unperturbed hamiltonian. In addi:ion we have used an uncoupled perturbation procedure rather than the more
the canoil-
the localized ones and various loca.Iization criteria have been used, see, e.g., refs. [2,7,8]. In practice the
different criteria all lead to very similar functions {w,). For the moment we shall assume t!!at what slight differences there are have no significant effect on the bond properties.
’
contributions a.ltiou& it k to be understood that it can refer to contributions from tiner shells as, for example, when the w; represents the orbital for the 1s electrons localized about a crrcbon zt0.m.
We use the term bond
581
Volume 22, number 3 accurate [ 121).
coupled
CHEMICAL
procedure
It is not difficult
(for
to write
a discussion, down
these
see ref. correct-
ing
terms but we shall not do so here since we believe they will make only minor corrections to our results and will almost certainly not lead to any qualitative changes.
4. Frequencydependent Waals interaction
perturbations
and van der
The type of approach outlined in the previous section can also be applied if the perturbation is time dependent. In particular, if the perturbation is the frequency-dependent one: cW= 11,exp(-iwr)
+ lvt exp(itijr),
=-W$o,
+ ($0,
then
the perturbation
be used
to fiid
procedure
the interaction
described energy.
earlier
After
1973 can
some
manipulation, the details of which will be given elsewhere, the interaction can be expressed, as a fist approximation, as the sum of all pairwise interactions between a localized bond in A and a localized bond in B. Apart from electrostatic type terms, the most irnportant contribution arises from a van der Waals type of interaction between pairs of bonds which has the usual form -dik/R6, where R is the distance between the centroids of charge in the two bonds and d,? is
the van der Waals coefficient. After averamg over orientations, this can be expressed in terms of the mean values Ej(iw) and Ek (iw) of the frequency dependent polarizability tensor d,l = (3/r;) J
all
W$+).
(1%
ff W and W+ hnve the separable form of eq. (7), then can be written as the sum of bond contributions,
CX(W)
(Yi(i(d) ‘Yk(io) dw .
0
5. Example.
(H;)-Bg+iiw)$+=-IY++~
(14) since the quantity of interest, e.g., components of the frequency-dependent polarizability n-ill be given by, QI(w) = ($0, w++_j
15 October
LETTERS
(13)
where Wand IV+ are hermitian conjugate operators independent of time, it is useful to introduce spatial functions $_ and $+ satisfying (H&!+fiw)$_
PHYSICS
The CH bond in methane
As an example we have taken the methane wavefunction of Palke and Lipscomb [ 131. The lowest canonical Hartree-Fock orbital is already well. localized and we have assumed it to be the carbon inner shell Is-like orbital ‘vl. The remaining four canoni
orbitals
have been
transformed
into
the
where w,,: and \vlT satisfy one-electron equations analogous to (8). Again there wil! be correcting terms arising from the fact that Hb is not the true zero-order molecular hamiltonian. Eq. (15) is meaningful even if w is complex. In particular it can be used to find the frequency-dependent bond polarizability at imaginary frequencies q(iw) by replacing w with iw. The quantity 9(iw>
four equivalent sp3 hybrid orbit&, rvZ, lv3, rv4, w5, i.e., the four CH bond orbitals. The components of the static polarizability tensor for the localized orbit& were computed using the variational-perturbation method (see, for example, Karplus and Kolker [ 141) with a two-term trial function. Taking the z axis along the bond, the components along the principal axes are given in table 1. Notice that, as is to be expected, the contribution from the inner shell localized orbital is almost zero. A similar calculation was made for the frequency dependent case except that only a one-term trial function was used. The orientationally-averaged contribu. tions to djk are shown in table 1. These results can be compared with other theoret-
is useful
ical and empirical
a(w) = with T(W) = 2{9,
z W,T’ + crvi, ZlVi+)) ,
for computing
the van der Waals
(17)
coefficient
estimates
and this is done
in table
between bonds. If two molecules A and B which are built up of
As cui be seen the agreement is quite reasonable. We believe these preliminary calculations show
localized bonds interact with each-other at a sufficient separation that exchange of electrons can be neglected,
that the procedure is a useful one for theoretically computing properties of localized bonds. We hope in
582
1.
Volume
22, number
3
Static
CHEMICAL
PHYSICS
polarizabilities
Table 1 and dispersion
15 October
LETTERS
cncrgies
1973
a)
Variationalperturbation method A.
Empirical
Polarizobilitie~
C-H
bond mean parallel transverse
cxbon CH4
C-H C-H
iraer
0.0086
shell
16.73 energies
units.
17.55c)
dJ 9.24 0.0308
-C-H - C is.
C is. - C is. - CH4 (34 are in atomic
4.38 b) 5.33 3.91
4.18 6.03 3.26
molcculc
B. Dispersion
a) AU values
Semiempirical
12.20
0.00061 148 b) Ref.
195 e)
[ 11.c) Ref. ]lS]. d, Coefficients
future to apply it to other types of bonds and other properties.
References
c)
of -Rm6.
e, Ref.
[16].
[7] S.F. Boys, Rev. Mod. Phys. 32 (1960) 300. [S] ,D. Peters, J. Chem. Sac. (196;) 2003,20:5,4017. [9] A.T. Amos and J. Musher, J. Chcm. Phys. 53 (1971) [IO] [I 1)
2380. S. Diner. J.P. Malricu Actn 13 (1969) 1.
and P. Clavcrie.
Theoret.
Chin-t.
5.0. Hirschfcldcr, W.B. Brown and S.T. Epstein, Acivan. Quantum Chcm. i (1964) 256.
Denbigh, Trans. Faraday Sot. 36 (1940) 936. [ 21 W. England, L.S. Salmon and K. Ruedenberg, Topics in current chemistry, Vol. 23 (Springer, Berlin, 197 1).
[ i2] A.T. Amos nnJ J. Slushcr, hlol. Phys. 13 (1967) 509. [ 131 W.E. Palkc and W.N. Lipscomb, J. Am. Chem. Sot. 88
[3] M.S. Gordon and W. England, J. Am. Chem. Sot. 94 (1972) 5168.
[ 141 hl. Karplus
[ 11 KG.
[4] [5] [6]
H.F. Hameka, J. Chem. Phys. 34 (1961) 1996. A.T. Amos and J. Musher, J. Chem. Phys. 49 (1968) 2158. G.G. Hall and J.E. Lennard-Jones, Proc. Roy. Sot. A202 (19.50) 155.
(1966)
2384. and H.J. Kolkcr,
J. Chem.
Phys.
38 (L963)
1263; 39 (1963) 2011. 1151 Landolt-Bornstein, Zahlcnwertc und Functionen, Vol. 1, part 3.2,‘eds. J. Bartel et al. (Springer. Berlin, 195 I) p- 354. [ 161 L. S&m, J. Chcm. Phys. 37 (1962) 2100.
583
15 October
CHEMICAL PHYSICS LETTERS
1973
SECOND-ORDERPROPERTIESOFBONDS A.T. AMOS and R.J. CRISPIN Depurtnrent
ojhlathemotics, The Unilerrity, Notringlram, UK
Received 26 July 1973
By defining a localized hamiltonian for a saturated molecule its second order molecular properties may be regarded as a sum of bond contributions. Static and dj-aamic pokuizabilities are calculated for the C-H bond in methane and the latter one used to calculate the van der Waals interaction between two CH4 molecules.
1. Introduction
For many years, and even before the advent of Mechanics, it has been known empirically that &properties of certain classes of molecules can be found by adding together contributions from the bonds and atoms in these molecules. Thus, to give an Quantum
example, the po!arizabiIity of a saturated hydrocarbon can be found by adding together empirical polarizabilities of the individual CC and CH bonds [l] I Naturally, as Quantum Mechanics began to be applied to chemical properties, there were many attempts to give a theoretical foundation to this empirical resuIt. Some of these attempts have been reasonably successful but none of them completely so, pstly because it is only recently that sufficiently accurate molecular wavefunctions have become available and partly because the. application of perturbation theory within the context of the Hartree-Fock scheme has
been fully understood only in the past few years. Now that these two difficulties have been removed it seems time for a reconsideration of the problem. Indeed, to some extent this has already begun with the fine work by Ruedenberg, England and their collaborators (for some references see ref. [2]). These, however, have in the main considered first-order properties, e.g., dip& moments [3] _Although there have been interesting discussions of second-order properties, e.g., polarizabilities and susceptibilities, the methods used have not been suitable for computing numerical values 530
of the second-order bond properties [4,5]. In this note we wish to suggest a method which can easily be used to define and to calculate values of the bond properties of CH and CC bonds in saturated hydrocarbons where reasonably accurate HartreeFock wavefunctions are available. As examples we use the method to fiid the static polarizabilities of the CH bond in methane and the van der Waals interaction
betyeen hvo methane-like CH bonds.
2. Bond orbitals Suppose we consider a molecule with localized bonds, e.g., a saturated hydrocarbon, and no delocalized electrons. L-eta0 be the ground-state Hartree-Fock wavefunction built up ofn doubly occupied molecular orbitals {cji), i= 1, ..., II, i.e.,
9’0 = I @la &P ..‘ $,a &jPI I
(1)
where the 4; satisfy hO@i= ei@i, 110being
t52
Normally
(2) Hartree-Fock the
hamiltonian.
{I#+-: are not themselves
localized
or-
bitals but they can bi transfarmed to a new and equivalent set {we), 01= 1, m-b,12,whic+re localized along the bonds and within the inner shells of atoms [6]. This transformation is the basis of .d theories of
Volume 22, number 3
bond properties *. Following the suggestion of Amos and Musher [V] which is analogous to the method used by Diner et al. [lo] we define a new one-electron operator 11; by n
C
hb =ho-
15 October
CHEMICAL PHYSICS LETTERS
where the changes in the orbit& are obtained from the equations (/zb - E;) NJ;.= (EJ - 2) Wj .
e4111wp>(waI
Pf”
in the Dirac notation, with ecro = (~,lh~l~~). hb and hr-,have the sameVirtual orbitals but the lowest JZ orbit& of Izb are the localized {w,) whereas the II lowest orbitals of ho are the canonical ones {gi]s Using the izb we can now define
1973
the k-electron
operator Hb by 2n (4)
and this has the property that
(8)
This equation can be solved by the sum-over-states method [ 111 if, in the original calculations ofqg, enou& basis orbit& are used to enable the virtual orbitals of 11, to form a reasonably complete set for the problem. As ;1 rule this will not be the case, and then variational-perturbation t,echniques [I l] can be used. Provided each rvi is chosen to be orthogonal to the unperturbed orbital rui the second-order change in the energy due to the perturbntion is @en by n E, =2~
{~Vi(Z
(9)
IWI_).
(5) Using (9) we can define the contribution from the ith bond orbital as
where we now write ‘k, in the transformed form XIJo= IWlcz
rvrp ... W,,LyW,pI
(6) E; = 2 (W&ZI\+,
and E, = 2 5
(10)
with the property that the total second-order energy will be the sum of these bond contributions ‘, i.e., Ei .
i=l
3. Perturbation
theory
Normally the second-order property is defied
Now we apply uncoupled perturbation theory. Let the perturbation be represented by the sum of one. electron operators, e.g., 2n
LyI.=-4(Wi(Z(WI).
w-gz(i).
(7)
The first-order wavefunction is given by
* There are many ical orbit& to
as -2E,, e.g., the components of the po!arizabWy or susceptibility tensor. If crj is the contribution from the ith bond-orbital to the second-order property, then
schemes
for transforminS
from
02)
Clearly there are correcting terms to be added to this expression. These arise essentially because ‘IX0is the Hartree-Fock wavefunction and not the correct onewith the consequence that HA is not the proper unperturbed hamiltonian. In addi:ion we have used an uncoupled perturbation procedure rather than the more
the canoil-
the localized ones and various loca.Iization criteria have been used, see, e.g., refs. [2,7,8]. In practice the
different criteria all lead to very similar functions {w,). For the moment we shall assume t!!at what slight differences there are have no significant effect on the bond properties.
’
contributions a.ltiou& it k to be understood that it can refer to contributions from tiner shells as, for example, when the w; represents the orbital for the 1s electrons localized about a crrcbon zt0.m.
We use the term bond
581
Volume 22, number 3 accurate [ 121).
coupled
CHEMICAL
procedure
It is not difficult
(for
to write
a discussion, down
these
see ref. correct-
ing
terms but we shall not do so here since we believe they will make only minor corrections to our results and will almost certainly not lead to any qualitative changes.
4. Frequencydependent Waals interaction
perturbations
and van der
The type of approach outlined in the previous section can also be applied if the perturbation is time dependent. In particular, if the perturbation is the frequency-dependent one: cW= 11,exp(-iwr)
+ lvt exp(itijr),
=-W$o,
+ ($0,
then
the perturbation
be used
to fiid
procedure
the interaction
described energy.
earlier
After
1973 can
some
manipulation, the details of which will be given elsewhere, the interaction can be expressed, as a fist approximation, as the sum of all pairwise interactions between a localized bond in A and a localized bond in B. Apart from electrostatic type terms, the most irnportant contribution arises from a van der Waals type of interaction between pairs of bonds which has the usual form -dik/R6, where R is the distance between the centroids of charge in the two bonds and d,? is
the van der Waals coefficient. After averamg over orientations, this can be expressed in terms of the mean values Ej(iw) and Ek (iw) of the frequency dependent polarizability tensor d,l = (3/r;) J
all
W$+).
(1%
ff W and W+ hnve the separable form of eq. (7), then can be written as the sum of bond contributions,
CX(W)
(Yi(i(d) ‘Yk(io) dw .
0
5. Example.
(H;)-Bg+iiw)$+=-IY++~
(14) since the quantity of interest, e.g., components of the frequency-dependent polarizability n-ill be given by, QI(w) = ($0, w++_j
15 October
LETTERS
(13)
where Wand IV+ are hermitian conjugate operators independent of time, it is useful to introduce spatial functions $_ and $+ satisfying (H&!+fiw)$_
PHYSICS
The CH bond in methane
As an example we have taken the methane wavefunction of Palke and Lipscomb [ 131. The lowest canonical Hartree-Fock orbital is already well. localized and we have assumed it to be the carbon inner shell Is-like orbital ‘vl. The remaining four canoni
orbitals
have been
transformed
into
the
where w,,: and \vlT satisfy one-electron equations analogous to (8). Again there wil! be correcting terms arising from the fact that Hb is not the true zero-order molecular hamiltonian. Eq. (15) is meaningful even if w is complex. In particular it can be used to find the frequency-dependent bond polarizability at imaginary frequencies q(iw) by replacing w with iw. The quantity 9(iw>
four equivalent sp3 hybrid orbit&, rvZ, lv3, rv4, w5, i.e., the four CH bond orbitals. The components of the static polarizability tensor for the localized orbit& were computed using the variational-perturbation method (see, for example, Karplus and Kolker [ 141) with a two-term trial function. Taking the z axis along the bond, the components along the principal axes are given in table 1. Notice that, as is to be expected, the contribution from the inner shell localized orbital is almost zero. A similar calculation was made for the frequency dependent case except that only a one-term trial function was used. The orientationally-averaged contribu. tions to djk are shown in table 1. These results can be compared with other theoret-
is useful
ical and empirical
a(w) = with T(W) = 2{9,
z W,T’ + crvi, ZlVi+)) ,
for computing
the van der Waals
(17)
coefficient
estimates
and this is done
in table
between bonds. If two molecules A and B which are built up of
As cui be seen the agreement is quite reasonable. We believe these preliminary calculations show
localized bonds interact with each-other at a sufficient separation that exchange of electrons can be neglected,
that the procedure is a useful one for theoretically computing properties of localized bonds. We hope in
582
1.
Volume
22, number
3
Static
CHEMICAL
PHYSICS
polarizabilities
Table 1 and dispersion
15 October
LETTERS
cncrgies
1973
a)
Variationalperturbation method A.
Empirical
Polarizobilitie~
C-H
bond mean parallel transverse
cxbon CH4
C-H C-H
iraer
0.0086
shell
16.73 energies
units.
17.55c)
dJ 9.24 0.0308
-C-H - C is.
C is. - C is. - CH4 (34 are in atomic
4.38 b) 5.33 3.91
4.18 6.03 3.26
molcculc
B. Dispersion
a) AU values
Semiempirical
12.20
0.00061 148 b) Ref.
195 e)
[ 11.c) Ref. ]lS]. d, Coefficients
future to apply it to other types of bonds and other properties.
References
c)
of -Rm6.
e, Ref.
[16].
[7] S.F. Boys, Rev. Mod. Phys. 32 (1960) 300. [S] ,D. Peters, J. Chem. Sac. (196;) 2003,20:5,4017. [9] A.T. Amos and J. Musher, J. Chcm. Phys. 53 (1971) [IO] [I 1)
2380. S. Diner. J.P. Malricu Actn 13 (1969) 1.
and P. Clavcrie.
Theoret.
Chin-t.
5.0. Hirschfcldcr, W.B. Brown and S.T. Epstein, Acivan. Quantum Chcm. i (1964) 256.
Denbigh, Trans. Faraday Sot. 36 (1940) 936. [ 21 W. England, L.S. Salmon and K. Ruedenberg, Topics in current chemistry, Vol. 23 (Springer, Berlin, 197 1).
[ i2] A.T. Amos nnJ J. Slushcr, hlol. Phys. 13 (1967) 509. [ 131 W.E. Palkc and W.N. Lipscomb, J. Am. Chem. Sot. 88
[3] M.S. Gordon and W. England, J. Am. Chem. Sot. 94 (1972) 5168.
[ 141 hl. Karplus
[ 11 KG.
[4] [5] [6]
H.F. Hameka, J. Chem. Phys. 34 (1961) 1996. A.T. Amos and J. Musher, J. Chem. Phys. 49 (1968) 2158. G.G. Hall and J.E. Lennard-Jones, Proc. Roy. Sot. A202 (19.50) 155.
(1966)
2384. and H.J. Kolkcr,
J. Chem.
Phys.
38 (L963)
1263; 39 (1963) 2011. 1151 Landolt-Bornstein, Zahlcnwertc und Functionen, Vol. 1, part 3.2,‘eds. J. Bartel et al. (Springer. Berlin, 195 I) p- 354. [ 161 L. S&m, J. Chcm. Phys. 37 (1962) 2100.
583