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Electric polarizability of polyatomic molecules
CHZMICAL
PHYSICS LETTER5
ELECTRIC
1 (1367) 242-215.
NORTH -HOLLAND
POLARIZABILITY G. P. ARRIGRINI,
PUBLISHING
OF POLYATOMIC _%¶. MAESTRO
Electric polarizabillties sf some scheme. by using LCAO SCF MO’s
polyatomic
1. INTRODUCTION In recent years, some results have been ob . tained in calculating molecular observables, like electric polar&ability and magnetic susceptibility, corresponding to one-electron perturbation terms in the Hamiltonian, within the HF scheme [l-33. Most cf these calculations were concerned with diatomic systems, for which, by chcosing suitable and extended basis sets, it was possible to obtain good agreement with experimental data. For pclyatomic molecules, however, as far as we know, there are only a few calculations (SOC z-g. ref. [3], where a one-center basis set is employed). In this letter, the results for the electric polarizability of some closed-shell polyatomic molecules, calculated within the HF scheme and employing multicenter basis sets, are reported.
2. THEORY
In principle, t!le electric polarizability could be calculated, in the HF scheme, by evaluating the HF energy for several values of the applied electric field, and, afterwards, computing the second order coefficient of the energy as a function of the field! [4]. By this direct method a sufficient precision is seldom attainable, especially when the first derivatives of the energy are not zero; moreover, the method is expensive, partictilar!y for off-diagonaI components. We have only employed this method as a test to control the reliability of results in some simple cases. A different approach involves the solution of the system of perturbed HF equations, following the lines exposed by several authors [1,3 $1.
Ital_s
13 June 1967
molecules
\ave functions.
MOLECULES
and R. MGCCIA
Centro di Chimica Teon’ca de1 C.N.R. Istituto di Cfzimica Fisica dell’(lnir~ersitir. Pisa. Received
COMPANY _ AhSTERDAM
are esaluated.
wthm
the HF gerturbative
This method allows one to pick up the wanted coefficients of +-heenergy expansion; we briefly recall the main formulae. The first order HF equations are (Jo-$)&$)
+ (@)+G(U$)),~
= 0. I,(p) (1)
wherefO is the unperturbed RF operator, the “external” perturbation, and, for real @) (50 that 0$?) are real too)
is the perturbation induced in the HF electronic field. In order to solve eq. (1) by an expansion technique, it is preferable to avoid working with matrix elements upon molecular orbitals; at this end, we express all the working formulae by integrals upon the original atomic basis functions {x). This way of proceeding appears particularly compact and time-saving. Be
It can be shown [3] that the unknown coefficient C$’
may be written:
ELECTRLCPOIARIZABLLITYOF
C!yl 2
UrlOCC[
g
d”
.-u
3. RESULTS
(&)+&))C?]
*/ c.;
9-c
.
(2)
The solution is attained by an iterative procedure, because G(3) is defined in terms of the unC(o); the process usually will start with gkr;e;:‘,,@)
ha b ve een obtainad, the observables evaluable by
are di.rect$ E(h)
= 2A~PR\$)&$
= 2A~P~~~$
(3)
where I@) is the second derivative of the HF energy with respect to the perturbation parameters S ,Y. The electric polarizability tensor (Yggl is so defimxi o!gg~ = -,$&$), c being the g-component *this case I@) = g.
&,g’
= X¶v, 2)
of the electric
PGLYATOMICMOLECULES
field e.
213
AND DISCUSSIONS
Tables l-3 collect the electric poIarizabiiity components for H20, NH3, CQ and H202 molecules, for different SCF basis sets. Tale 1 shows the results for H20 relative to sever. different basis sets; the numwr of atomic functions is reported in column 1, md the basis is qc:ified in column 2. TIie notation aD(<)MdD(<)] means that all three [five] functions associated with the same < value, centered at the atom D, are included. For H20, the t axis was chosen along the two-fold axis C2, while the molecular plane> xz: experimental tiues for OH distance and HOH angle were assumed. One can see that the results for minimal basis sets are very poor, and ‘the agreement with experiment is only slightly improved when going to double-zeta sets. As to be expected, these results show a marked anisotropy of the polarizability, but this effect is clearly enhanced by the limited basis which can be unbalanced for this
Table 1 Results forH20 basis
cm3) Em,l(a.u.) (Debye) (10z5cm3) (LO2'-
Basis functions
functions
L 1so(7.658):2sO(2.3=161); 7
(10-35cm3) (1025,,3)
-75.66139[6] 1.631[6] 11.0793
;
1+
4.1874
5.0989
-76.00505[6) 2.646[6]
9.9094
1.8245
5.8679
5.P6C9
8.4039
1.64X
1.6941
4.9817
11.4974
11.3590
x
PpO(2.22625):lsH(l.O0) 1so(7.06227):2so(l.62705); 2po(1.65372) 14
II
1s'(10.1095);2!3~(2.62158); 2pj(3.68127);LsH(Z.L);ls~(l.-I) ls~(7.0622;);2~~(1.62505~;
14
2po(L.6537P) III 1a~(10.1085);2s~(2.6P1Sa);
-76.005%[6] 2.638[6]
190(7.45);%0(2.17):2p0(2.15); 3SO(1.5) 27a)
l-V1sj(13.0);2s~(3.50);3p0(1.5)
-76.0374 [6] 1.961[6] 12.4664
10.1133
-76.0263 [6] 2.359[6] 10.9773
5.4800
12.2619
10.0631
11.0849
11.1233
-76.0374 [6] 1.961[6] 11.0630
8.9696
11.4974
LO.5LOU
3dO(1.5);lsR(1.0):ls;I(1.5); qi(1.1) 32 23 26
v as IV except the five3d0 VIasIV
exceptthe2pH
VB as JX except the (d&O E.xperimental
-76.0346 [S] 1.912[6]
-76.465 [7] 1.850[7]
a) Only 2P,. 2p, on the hydrogen atoms.
lO.-vll9
-
8.9664
14.56[8]
G. P. ARRIGHINI.
244
M. MAESTRO Table
Results NIf3@ No. of basis
Basis
functions
functions
I
a
2
for NH3.
NH3 II
and R. MOCCIA
CH4.
H202
NH3 III
16
CH4 I
32
Hz02
ia
9
30
lsN(6.67457)
lsN(6.11663)
16N(
l~~(5.716)
l~~(5.2309)
lso(
3sN(l.9426)
l&8.93843)
lsN(19.8389)
2sC(1.625)
IsC(7.96897)
lgJ(13.0)
2pN(l .S426)
2q.JI.39327)
2sN(
2pc(1.625)
2SC(1.16782)
2so(1.62705)
2&(2.22157)
2Sfr(1.393)
2pN(1.50585)
2sFy(5’.033)
lSH(1.19545)
6.4595)
CH4 II
2.5)
2p$3.26741)
2pN(2 .3)
lSH(1.40)
2pk(l-3)
lSH(1.28)
3dH(1.8)
l&(1.65)
7.658)
2sC(l.82031)
2s~(2_6215S)
2pC(1.25572)
2pO(1.65372)
ZpC(2 -72625)
2pb(3.68127)
lSH(1.4)
lSH(1.4)
lsH(1.65)
2PH( l-4)
lsH(1.3) lSH(1.5) 2PH(l.4) lSC,P(l.O)b) Em01
Calc. Ex&
cr
Calc. Exp.
crrx
-56.00576 [6] -56.573 fll] 1.783 1.48
161 1111
Exp.
8.4937 21.8
(141
Calc. Exp.
8.4937 21.8
@.zt
‘Jalc. EXP.
1.7306 34.2
r141
Q1
Caic. Exp.
6.2393 22.6
r141
CalC.
-56.16746 161 -56.573 ::I:”
r6’
11.3021
-56.18604 [9] -56.573 ;4;5
-40.11287 [S] -40.522 [ 121
1%
0 0 8.1767 26.0
15.1705
-40.15455[6] -40.522
-150.8347[10] -151.6017[13]
0 0
2.156 2.26
14.5824
14.4871
t141
11.3021
15.1705
8.1767 26.0
14.5824
6.4420
5.2860
11.2435
8.1767 26.0
14.5824
4.4945
9.2967
13.8615
8.1767 26.0
14.5824
8 -4745
a) The geometry is the experimental one scaled by a factor 1.0033. b) Located upon the estimated centroid of the lone pair (0.7 au. from
observable. It is noteworthy that cases II and III, which only differ with respect to the functions centered
at the hydrogen atoms, and whose energy and
dipole moment are almost equal, give rise to markedly different polarizability components. The last four cases show the sensibility of the results upon the choice of the basis. For rather extended sets, one may see that d-functions on the oxygen atom are in general effective in imFroving the results; in particular, the d, function, which, for symmetry reasons, Ys not involved in the unperturbed case, appears, as expected [3], fairly important in increasing the polarizability.
[lo] [13]
N).
In table 2 are collected the results for other molecules. In aU cases, except for NH3 I (see footnote at table 2), the adopted geometrical configuration is that of equilibrium. The coordinate framework for NIX3 is with the 2 axis along the threefold axis, while, of course, for CH,qthe orientation is immaterial. As far as H202 is concerned, the principal axes of polarizability are specified by the angle 40in fig. 1, the z axis being now coincident with the C2 one. The only experimental data we have found in the literature are relative to NH3 and Cq; one can see that minimal basis and double-zeta set results, analogously to the H20 case, are stiIl in bad- agreement with them, even though some
ELEC'IRIC Table
POLARIZARILITYOF
POLYATOMIC
MOLECULES
245
3
Results when the modification
of the HF field is
neglected.
Hz0 H20 H2G H2G H2G H2G H20 NH3 NH3 NH2 CH4 CH4 H202
I
II III IV V VI VII I II III I II
Qxx
OYS
R?z
7.3680 6.4841 6.9702 10.944 8.6496 10.786
=O
3.9595 4.5980 3.9573 10.185 8.3172 9.8107 10.185 1.4268 4.1652 10.413 6.7185 11.832 3.9318
1.2523 1.3225 8.7152 3.6638 8.7016
8.8105
7.1436
6.3301 6.6296 13.547
6.3301 8.6296 13.547 6.7185 11.832 6.0962
6.7185 11.832 11.324
3.7758 4.1115 4.0833 9.9481 6.9435 9.7661 8.7130 4.6957 7.1415 12.502 6.7185 11.832 7.1173
I
relative improvement is evident when going to thedouble-zetaones. Itmustbe noted, however, that, particularly in the CH4 case, the result for
the double-zeta set is extremely good from the energy viewpoint. As already seen for H20 IV, the extended basis set for NH3 leads to a markedly improved, although still unsatisfactory, result. In order to compare these results with those relative to very extended one-center basis sets, see [3]. The reported cases show how critical the results are to the virtual orbitals available. For H202wepresumethatthe energy calculated bythe chosen basis is notfarfrom the estimated HF limit[10]. Since the result for the dipole moment is good, one would expected a less
marked anisotropy of the polarizability components. It is interesting to note that the used iterative process was not convergent when applied to a H202 minimal basis set. In table 3 we list the results obtained when the G(p) operator in eq. (2) is cancelled. One may conclude that the contributions of the perturbation of the HF electronic field is rather important. Among other things, in H202 it gives also rise to a slightly different orientation of the principal axes of polarizability; as a matter of fact, the cp angle (defined in fig. 1) is 50 when G(p) is dropped to be compared with cp = 14029’ when included. *****
Fig.
FIG1
1.
The authors are very grateful to Prof. 0. Salvetti and all others who made available ail necessary integrals and molecular wave functions prior to publication. REFERENCES [I]
R.M.Stevens, R-M. Pitzer and W.N.Lipscomb, J. Chem.Phys. 38 (1963) 550. [2] R.M.Stevens and W.N.Lipscomb, J.Chem.Phys. 42 (1965) 3666. [3l R.Moccia, Theor.Chim.Acta, in press. [4) H.D.Cohen and C.C.J.Roothaan, J.Chem.Phys. 43 (1965) 534. [5] R.McWeeny, Phys.Rev. 126 (I962j 1625. [6] O.Salvetti, private communication. [7] J.W.Moscowitz and M.C. Harrison, J.Chem.Phys. 43 (1965) 3550. [8] R.SPnger, Physik.2. 31 (1930) 306. 191R.Moccia, to be published. [ 101 C .Guidotti, M. Maestro, R. Moccia and O.SaLvetti. to be published. [ll] U.Kaldor and I.Shavitt. J.Chem.Phys. 45 (1966) 888. [=I BI. .Yrauss, J. Chem. Phys. 38 (1963) 564. 44 (1966) t131 U.Kaldor and I.Shavitt, J.Chem.Phys. 1826. Zahlenwerte und Funktionen. I141 Landolt-Boernstein: vol. I. part 3 (Springer Verlag) p. 511.
PHYSICS LETTER5
ELECTRIC
1 (1367) 242-215.
NORTH -HOLLAND
POLARIZABILITY G. P. ARRIGRINI,
PUBLISHING
OF POLYATOMIC _%¶. MAESTRO
Electric polarizabillties sf some scheme. by using LCAO SCF MO’s
polyatomic
1. INTRODUCTION In recent years, some results have been ob . tained in calculating molecular observables, like electric polar&ability and magnetic susceptibility, corresponding to one-electron perturbation terms in the Hamiltonian, within the HF scheme [l-33. Most cf these calculations were concerned with diatomic systems, for which, by chcosing suitable and extended basis sets, it was possible to obtain good agreement with experimental data. For pclyatomic molecules, however, as far as we know, there are only a few calculations (SOC z-g. ref. [3], where a one-center basis set is employed). In this letter, the results for the electric polarizability of some closed-shell polyatomic molecules, calculated within the HF scheme and employing multicenter basis sets, are reported.
2. THEORY
In principle, t!le electric polarizability could be calculated, in the HF scheme, by evaluating the HF energy for several values of the applied electric field, and, afterwards, computing the second order coefficient of the energy as a function of the field! [4]. By this direct method a sufficient precision is seldom attainable, especially when the first derivatives of the energy are not zero; moreover, the method is expensive, partictilar!y for off-diagonaI components. We have only employed this method as a test to control the reliability of results in some simple cases. A different approach involves the solution of the system of perturbed HF equations, following the lines exposed by several authors [1,3 $1.
Ital_s
13 June 1967
molecules
\ave functions.
MOLECULES
and R. MGCCIA
Centro di Chimica Teon’ca de1 C.N.R. Istituto di Cfzimica Fisica dell’(lnir~ersitir. Pisa. Received
COMPANY _ AhSTERDAM
are esaluated.
wthm
the HF gerturbative
This method allows one to pick up the wanted coefficients of +-heenergy expansion; we briefly recall the main formulae. The first order HF equations are (Jo-$)&$)
+ (@)+G(U$)),~
= 0. I,(p) (1)
wherefO is the unperturbed RF operator, the “external” perturbation, and, for real @) (50 that 0$?) are real too)
is the perturbation induced in the HF electronic field. In order to solve eq. (1) by an expansion technique, it is preferable to avoid working with matrix elements upon molecular orbitals; at this end, we express all the working formulae by integrals upon the original atomic basis functions {x). This way of proceeding appears particularly compact and time-saving. Be
It can be shown [3] that the unknown coefficient C$’
may be written:
ELECTRLCPOIARIZABLLITYOF
C!yl 2
UrlOCC[
g
d”
.-u
3. RESULTS
(&)+&))C?]
*/ c.;
9-c
.
(2)
The solution is attained by an iterative procedure, because G(3) is defined in terms of the unC(o); the process usually will start with gkr;e;:‘,,@)
ha b ve een obtainad, the observables evaluable by
are di.rect$ E(h)
= 2A~PR\$)&$
= 2A~P~~~$
(3)
where I@) is the second derivative of the HF energy with respect to the perturbation parameters S ,Y. The electric polarizability tensor (Yggl is so defimxi o!gg~ = -,$&$), c being the g-component *this case I@) = g.
&,g’
= X¶v, 2)
of the electric
PGLYATOMICMOLECULES
field e.
213
AND DISCUSSIONS
Tables l-3 collect the electric poIarizabiiity components for H20, NH3, CQ and H202 molecules, for different SCF basis sets. Tale 1 shows the results for H20 relative to sever. different basis sets; the numwr of atomic functions is reported in column 1, md the basis is qc:ified in column 2. TIie notation aD(<)MdD(<)] means that all three [five] functions associated with the same < value, centered at the atom D, are included. For H20, the t axis was chosen along the two-fold axis C2, while the molecular plane> xz: experimental tiues for OH distance and HOH angle were assumed. One can see that the results for minimal basis sets are very poor, and ‘the agreement with experiment is only slightly improved when going to double-zeta sets. As to be expected, these results show a marked anisotropy of the polarizability, but this effect is clearly enhanced by the limited basis which can be unbalanced for this
Table 1 Results forH20 basis
cm3) Em,l(a.u.) (Debye) (10z5cm3) (LO2'-
Basis functions
functions
L 1so(7.658):2sO(2.3=161); 7
(10-35cm3) (1025,,3)
-75.66139[6] 1.631[6] 11.0793
;
1+
4.1874
5.0989
-76.00505[6) 2.646[6]
9.9094
1.8245
5.8679
5.P6C9
8.4039
1.64X
1.6941
4.9817
11.4974
11.3590
x
PpO(2.22625):lsH(l.O0) 1so(7.06227):2so(l.62705); 2po(1.65372) 14
II
1s'(10.1095);2!3~(2.62158); 2pj(3.68127);LsH(Z.L);ls~(l.-I) ls~(7.0622;);2~~(1.62505~;
14
2po(L.6537P) III 1a~(10.1085);2s~(2.6P1Sa);
-76.005%[6] 2.638[6]
190(7.45);%0(2.17):2p0(2.15); 3SO(1.5) 27a)
l-V1sj(13.0);2s~(3.50);3p0(1.5)
-76.0374 [6] 1.961[6] 12.4664
10.1133
-76.0263 [6] 2.359[6] 10.9773
5.4800
12.2619
10.0631
11.0849
11.1233
-76.0374 [6] 1.961[6] 11.0630
8.9696
11.4974
LO.5LOU
3dO(1.5);lsR(1.0):ls;I(1.5); qi(1.1) 32 23 26
v as IV except the five3d0 VIasIV
exceptthe2pH
VB as JX except the (d&O E.xperimental
-76.0346 [S] 1.912[6]
-76.465 [7] 1.850[7]
a) Only 2P,. 2p, on the hydrogen atoms.
lO.-vll9
-
8.9664
14.56[8]
G. P. ARRIGHINI.
244
M. MAESTRO Table
Results NIf3@ No. of basis
Basis
functions
functions
I
a
2
for NH3.
NH3 II
and R. MOCCIA
CH4.
H202
NH3 III
16
CH4 I
32
Hz02
ia
9
30
lsN(6.67457)
lsN(6.11663)
16N(
l~~(5.716)
l~~(5.2309)
lso(
3sN(l.9426)
l&8.93843)
lsN(19.8389)
2sC(1.625)
IsC(7.96897)
lgJ(13.0)
2pN(l .S426)
2q.JI.39327)
2sN(
2pc(1.625)
2SC(1.16782)
2so(1.62705)
2&(2.22157)
2Sfr(1.393)
2pN(1.50585)
2sFy(5’.033)
lSH(1.19545)
6.4595)
CH4 II
2.5)
2p$3.26741)
2pN(2 .3)
lSH(1.40)
2pk(l-3)
lSH(1.28)
3dH(1.8)
l&(1.65)
7.658)
2sC(l.82031)
2s~(2_6215S)
2pC(1.25572)
2pO(1.65372)
ZpC(2 -72625)
2pb(3.68127)
lSH(1.4)
lSH(1.4)
lsH(1.65)
2PH( l-4)
lsH(1.3) lSH(1.5) 2PH(l.4) lSC,P(l.O)b) Em01
Calc. Ex&
cr
Calc. Exp.
crrx
-56.00576 [6] -56.573 fll] 1.783 1.48
161 1111
Exp.
8.4937 21.8
(141
Calc. Exp.
8.4937 21.8
@.zt
‘Jalc. EXP.
1.7306 34.2
r141
Q1
Caic. Exp.
6.2393 22.6
r141
CalC.
-56.16746 161 -56.573 ::I:”
r6’
11.3021
-56.18604 [9] -56.573 ;4;5
-40.11287 [S] -40.522 [ 121
1%
0 0 8.1767 26.0
15.1705
-40.15455[6] -40.522
-150.8347[10] -151.6017[13]
0 0
2.156 2.26
14.5824
14.4871
t141
11.3021
15.1705
8.1767 26.0
14.5824
6.4420
5.2860
11.2435
8.1767 26.0
14.5824
4.4945
9.2967
13.8615
8.1767 26.0
14.5824
8 -4745
a) The geometry is the experimental one scaled by a factor 1.0033. b) Located upon the estimated centroid of the lone pair (0.7 au. from
observable. It is noteworthy that cases II and III, which only differ with respect to the functions centered
at the hydrogen atoms, and whose energy and
dipole moment are almost equal, give rise to markedly different polarizability components. The last four cases show the sensibility of the results upon the choice of the basis. For rather extended sets, one may see that d-functions on the oxygen atom are in general effective in imFroving the results; in particular, the d, function, which, for symmetry reasons, Ys not involved in the unperturbed case, appears, as expected [3], fairly important in increasing the polarizability.
[lo] [13]
N).
In table 2 are collected the results for other molecules. In aU cases, except for NH3 I (see footnote at table 2), the adopted geometrical configuration is that of equilibrium. The coordinate framework for NIX3 is with the 2 axis along the threefold axis, while, of course, for CH,qthe orientation is immaterial. As far as H202 is concerned, the principal axes of polarizability are specified by the angle 40in fig. 1, the z axis being now coincident with the C2 one. The only experimental data we have found in the literature are relative to NH3 and Cq; one can see that minimal basis and double-zeta set results, analogously to the H20 case, are stiIl in bad- agreement with them, even though some
ELEC'IRIC Table
POLARIZARILITYOF
POLYATOMIC
MOLECULES
245
3
Results when the modification
of the HF field is
neglected.
Hz0 H20 H2G H2G H2G H2G H20 NH3 NH3 NH2 CH4 CH4 H202
I
II III IV V VI VII I II III I II
Qxx
OYS
R?z
7.3680 6.4841 6.9702 10.944 8.6496 10.786
=O
3.9595 4.5980 3.9573 10.185 8.3172 9.8107 10.185 1.4268 4.1652 10.413 6.7185 11.832 3.9318
1.2523 1.3225 8.7152 3.6638 8.7016
8.8105
7.1436
6.3301 6.6296 13.547
6.3301 8.6296 13.547 6.7185 11.832 6.0962
6.7185 11.832 11.324
3.7758 4.1115 4.0833 9.9481 6.9435 9.7661 8.7130 4.6957 7.1415 12.502 6.7185 11.832 7.1173
I
relative improvement is evident when going to thedouble-zetaones. Itmustbe noted, however, that, particularly in the CH4 case, the result for
the double-zeta set is extremely good from the energy viewpoint. As already seen for H20 IV, the extended basis set for NH3 leads to a markedly improved, although still unsatisfactory, result. In order to compare these results with those relative to very extended one-center basis sets, see [3]. The reported cases show how critical the results are to the virtual orbitals available. For H202wepresumethatthe energy calculated bythe chosen basis is notfarfrom the estimated HF limit[10]. Since the result for the dipole moment is good, one would expected a less
marked anisotropy of the polarizability components. It is interesting to note that the used iterative process was not convergent when applied to a H202 minimal basis set. In table 3 we list the results obtained when the G(p) operator in eq. (2) is cancelled. One may conclude that the contributions of the perturbation of the HF electronic field is rather important. Among other things, in H202 it gives also rise to a slightly different orientation of the principal axes of polarizability; as a matter of fact, the cp angle (defined in fig. 1) is 50 when G(p) is dropped to be compared with cp = 14029’ when included. *****
Fig.
FIG1
1.
The authors are very grateful to Prof. 0. Salvetti and all others who made available ail necessary integrals and molecular wave functions prior to publication. REFERENCES [I]
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