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Magnetic circular dichroism spectra of 3,5-dimethylcyclopenta [ef] heptalene
Spcctrochimica Acta, Vol.35A,pp.1147 to 1149 Pergamon Press Ltd., 1979. F’rinted in Northern Ireland
Magnetic circular dichroism spectra of 3,Slimethylcyclopenta
[ef] heptalene
H. YAMAGUCHI* and T. KOYANAGI Department of Chemistry, Faculty of Science, Kumamoto University, Kurokami, Kumamoto 860, Japan
Department
T. MURAOICA of Chemistry, Faculty of Education, Miyazaki University, Funatsuka, Miyazaki 880, Japan and
K. HAFNER Institut fiir Organische Chemie der Technischen Hochschule Darmstadt, D-61 Darmstadt, West Germany (Received
22 November 1978)
Abstract-Magnetic circular dichroism (MCD) and absorption spectra of 3,5-dimethylcyclopenta[ef]heptalene have been measured. The absorption spectra have been assigned by comparing the experimental results with the signs of Faraday B terms and transition energies computed by using the Pariser-Parr-Pople SCF-r-MO Cl method incorperated with the variable bond-length technique. INTRODUCI’ION The
electronic spectra of cyclopenta[ef]heptalene (I) have long been of interest since HAILER and SCHNEIDER [l] prepared the 3,5-dimethyl derivative of I in 1958-1959. From a theoretical point of view, the electronic spectra of I have been investigated by many authors [2-4]. EATON et al. [S] have shown that the observed emission of the 3,Sdimethyl derivative of I corresponds to anomalous emission due to a transition from second excited singlet. YAMAGUCHI et al. [6] have determined the dipole moments of the lower two singlet excited states of the 3,5dimethyl derivative of I from the frequency shift of the absorption and fluorescence spectra in various solvents and have pointed out that the direction of the dipole moments of both excited states is opposite to the direction of the ground-state dipole moment. Recently, magnetic circular dichroism (MCD) has been shown to be a useful tool for the assignment of absorption bands and for the investigation of the magnetic properties of the ground and excited states of molecules [7-91. The aim of this paper is to make the assignment of the absorption bands of the 3,5-dimethyl derivative of I by measuring the MCD spectra and by comparing the experimental results with the theoretical results for I obtained by use of the PARISER-PARR-POPLE
(PPP)
method [lo, 111 in combination bond-length technique [12].
hexane and by subsequent sublimation in and was spectroscopically pure, from comparison with published absorption spectra [5]. Cyclohexane was carefully purified according to the descriptions in Reference [13]. Absorption spectra were measured on a Hitachi EPS-3T recording spectrophotometer. MCD spectra were obtained with JASCO J-40 A and C recording spectropolarimeters equipped with a 15.2 kG electromagnet. All measurements were made on deaerated samples at room temperature. The Faraday B values were extracted from the MCD spectra by use of the formula: from
SCF-T-MO
mcuo
B =-(33.53)-l
[
Jtmd
([&/v)dv
where u was frequency in cm-’ and [e], was molar ellipticity per unit field in units of deg. I.m-’ mol-’ G-’ [8]. The MCD and absorption spectra were analyzed assuming G-type absorption bands with the method described by BADOZ et al. [14]. MO CALCULATION
We have calculated the electronic transition energies (AE), the dipole strengths (D), and the Faraday B values (B) of I by use of the PARISERPARR-P• PLE method [lo, 111 incorperated with the variable bond-length technique [12]. The 49
CI
with the variable
EXPERIMENTAL
The 3,5dimethyl derivative of I was prepared by one of the present authors (K. H.) [l]. This compound was carefully purified by recrystallization * To whom inquires should be directed.
Fig. 1. Numbering of carbon atoms and choice of axes. 1147
H. YAMAGUCHI,T. KOYANAGI, T. MURAOKA and K. HAFNER
1148 singly excited figuration
configurations
interactions.
electron
repulsion
evaluated
by
11.16
potential
and
(I,)
were
exponential
formula
[12].
moments
Faraday
B
employed values
the procedure evaluated
quantities ING [19],
described
and
the
respectively.
Two-
The
by use of the
magnetic
and elec-
evaluation
calculated
of the
according
to
by KAITO et al. [16]
and
operator
Liiwon+orthogonalized
atomic
[17], basis
were used for the computation of both [18]. According to CALDWELL and EYRthe
Faraday
B
terms
calculated
for
RFSULTS AND DISCUSSION The
absorption
and
MCD
spectra
of the
I
I
I
1
32
36
40
4L
a
noncentric molecule using an incomplete set of state functions were origin dependent. In this paper, origin was set at the midpoint between the centers of charges for the ground and lower five singlet excited states.
dimethyl
35
derivative
of I are shown in Figs. 2 and 3. As the MCD spectra observed in higher wavenumber region than 44.0 x lo3 cm-’ are rich of noises, we do not show the MCD spectra in Fig. 3. The calculated transition energies (AE), dipole strengths (D), and Faraday B values (B) for I are listed in Table 1 along with the experimental results for the 3,Sdimethyl derivative of I. The calculated transition energies are in fairly good agreement
[‘I, 0.0
40
KCM-’
Fig. 3. Magnetic circular dichroism spectrum (top) and absorption spectrum (bottom) of 3,5dimethylcyclopenta[efl-heptalene in the wavenumber region of 29.6-44.0x lo3 cm-tin
cyclohexane
at room temperature.
with the experimental transition energies. The agreement between the calculated and experimental values for D and B is not so good as expected. However, the predicted signs of the Faraday B terms extracted from the MCD spectra. The first absorption band exists in the wavenumber region of about 8.0~ lo3 to 20.0~ lo3 cm-‘. The MCD spectra of this band show a negative curve. The calculated transition energy for this band is 12.4~ lo3 cm-’ and the calculated Faraday B term for this transition shows the positive value (the sign of the Faraday B term is opposite to the sign of the MCD spectra). As the predicted transition energy and sign of the MCD spectra for this band are in good agreement with the experimental results for this band, the first band can be assigned to the ‘A, -+ ‘B2 transition. The MCD spectra appeared in the wavenumber region of 20.0-24.0 x lo3 cm-i show the positive and negative curves. The calculated transition energy and Faraday B term are 23.4 X lo3 cm-’ and positive, Table 1. Transition energies (AE), dipole and Faraday
12
16
20
2L
28
KCM’-’
Fig. 2. Magnetic circular dichroism spectrum (top) and absorption spectrum (bottom) of 3,5-dimethylcyclopenta[ef]-heptalene in the wavenumber region of about 8.0X lo3 to 29.6 x lo3 cm-l in cyclohexane at room temperature.
Type
AE*
lA, +lB,
12.4 23.4 28.1 30.9 36.1 36.9 40.0
IAl + ‘A, ‘Al+ ‘B, ‘A, + IA, IAl + rB, IA, -+‘A1 ‘A,+‘B,
*In lo3 cm-r. tin Debye’. Sin 1O-3 Bohr
Dt
(D)
Experimental B$
2.96 3.85 2.19 0.364 39.5 3.32 15.3 -8.21 41.3 -0.948 55.8 24.7 30.7 -22.9
magneton
strengths
B values (B)
Theoretical
Transition
0
23
- -2.0
affinas Ip =
in the
_
valence
taken
by use of a dipole velocity
respectively. functions
were
computed
were
respectively [lo]
and the electron
E, = 0.03 eV,
integrals
in con-
two-center
[ 151. The
atom
center-core tric
were
equations
for the carbon
eV
and
PARISER-PARR
NISHIMOTCA~ATAGA ity (E,)
one-
integrals
the
state ionization
were included
The
Debye’
AE*
Dt
BS
12.6 23.5 27.4 31.5 36.8 38.5 42.0
0.492 0.507 16.3 7.85 21.5 24.9 14.2
0.596 0.052 2.31 -0.639 -0.692 2.22 -2.19
cm-‘.
Magnetic circular dichroism spectra of 3$dimethylcyclopenta[ef]heptalene respectively. From these facts, it can be concluded that the band at 23.5 X lo3 cm-’ is ascribed to the second transition (IA,+ IAl). As pointed out by CALDWELL and EYRING [19], it is considered that the positive sign of the MCD spectra at 22.0X lo3 cm-’ comes from coupling with the vibrational modes different from the vibrational modes coupled at 23.5 x lo3 cm-‘. The MCD spectra have the negative curve in the wavenumber region of 24.0-29.3 x lo3 cm-l. The calculated transition energy (28.1 x lo3 cm-l) agrees with the experimental value (27.4 x lo3 cm-l). As the calculated Faraday B term shows the positive value, the absorption band presented in this wavenumber region can be assigned to the third transition (IAl -+ ‘BZ). The absorption and MCD spectra located in the wavenumber region of 29.5-36.0 x lo3 cm-’ have an abundance of vibrational structures. The MCD spectra indicate the positive peak at 3 1.5 X lo3 cm-’ and the negative curve in the wavenumber region of 32.0~ lo3 to 36.0~ lo3 cm-‘, respectively. As the calculated transition energy and Faraday B value are 30.9~ lo3 cm-’ and negative, respectively, the absorption band being at 31.5 x lo3 cm-’ can be assigned to the fourth transition (IAl + ‘A,). The appearance of the negative curve in the wavenumber region of 32.0-36.0 x lo3 cm-’ may be attributed to the coupling between the fourth transition and the vibrational modes different from the vibrational modes coupled with the fourth transition [19]. The transition energy and Faraday B value for the fifth transition are 36.1 x lo3 cm-’ and negative, respectively. As the MCD spectra take a positive peak at 36.8 x lo3 cm-‘, it can be expected that the fifth transition (‘A, + lBz) locates at 36.8X lo3 cm-‘. Since the calculated transition energy and Faraday B value are 36.9 x lo3 cm-’ and positive, respectively, the negative MCD peak appeared at 38.5 x lo3 cm-’ belongs to the sixth transition (‘A, + ‘A,). The positive MCD peak is observed at 42.0~
SA(A) 35/l&-c
1149
lo3 cm-‘. From the presence of this MCD peak, it can be considered that the seventh transition (‘A,+ ‘B2) exists in the wavenumber region of about 39.0 x lo3 to 44.0 X lo3 cm-‘. From the above assignments, it can be concluded that the absorption and MCD spectra of I are little affected by the introduction of the methyl groups at the 3 and 5 positions.
REFERENCES
PI K. HAFNER and J. SCHNEIDER,Agnew. Chem. 70, 702 (1958); Justus Liebigs Ann. Chem. 624, 37
(1959).
PI M. A. ALI and C. A. COULSON,Mol. Phys. 4, 65 (1961). [31 A. JULG and P. FRANCOIS,Theor. Chim. Acta 7, 101
(1966). [41 H. YAMAGUCHI, T. TERASAKA and T. NAKAJIMA, Theor. Chim. Acta 18, 255 (1970). [51 D. F. EATON, T. R. EVANS and P. A. LEERMARKERS, Mol. Photochem. 1, 347 (1969). [61 H. YAMAGUCHI,T. IKEDA and H. MAMETSUKA,Bull.
Chem. Sot. Japan 49, 1762 (1976). [71 A. D. BUCKINGHAMand P. J. STEPHENS,Ann. Rev.
Phys. Chem. 17, 399 (1966). PI P. N. SCHATZ and A. J. MCCAFFREY,Quart. Revs., Chem. Sot. 23, 552 (1969). [91 D. CALDWELL,J. M. THORNE and H. EYRING, Ann. Rev. Phys. Chem. 22, 259 (1971). [lOI R. PARISERand R. G. PARR, J. Chem. Phys. 21,446,
767 (1953). [ll] J. A. POPLE, Trans. Faraday Sot. 49, 1375 (1953). [12] H. YAMAGUCHI, T. NAKAJIMA and T. L. KUNII, 7’heor.Chim. Acta 12, 349 (1968). [13] J. A. RIDDICKand W. B. BUNGER, Organic Solvent, Vol. II, (edited by A. WEISSBERGER).Wiley, New York (1970). [14] J. BADOZ, M. BILLARDON,A. C. BOCCARA and B. BRIAT, Symp. Faraday Sot. 3, 27 (1969). r151 N. MATAGA and K. NISHIMOTO,Z. Phvsik. Chem. . a
13, 140 (1957). [16] A. KAITO, A. TAJIRIand M. HATANO, J. Am. Chem. sot. 97, 5059 (1975). [17] A. IMAMURA, T. HIRANO, C. NAGATA and T. TSURUTA, Bull. Chem. Sot. Japan. 45, 396 (1972). [18] P. 0. L~WDIN, J. Chem. Phys. 18, 365 (1950). [19] D. J. CALDWELLand H. EYRING, J. Chem. Phys. 58, 1149 (1973).
Magnetic circular dichroism spectra of 3,Slimethylcyclopenta
[ef] heptalene
H. YAMAGUCHI* and T. KOYANAGI Department of Chemistry, Faculty of Science, Kumamoto University, Kurokami, Kumamoto 860, Japan
Department
T. MURAOICA of Chemistry, Faculty of Education, Miyazaki University, Funatsuka, Miyazaki 880, Japan and
K. HAFNER Institut fiir Organische Chemie der Technischen Hochschule Darmstadt, D-61 Darmstadt, West Germany (Received
22 November 1978)
Abstract-Magnetic circular dichroism (MCD) and absorption spectra of 3,5-dimethylcyclopenta[ef]heptalene have been measured. The absorption spectra have been assigned by comparing the experimental results with the signs of Faraday B terms and transition energies computed by using the Pariser-Parr-Pople SCF-r-MO Cl method incorperated with the variable bond-length technique. INTRODUCI’ION The
electronic spectra of cyclopenta[ef]heptalene (I) have long been of interest since HAILER and SCHNEIDER [l] prepared the 3,5-dimethyl derivative of I in 1958-1959. From a theoretical point of view, the electronic spectra of I have been investigated by many authors [2-4]. EATON et al. [S] have shown that the observed emission of the 3,Sdimethyl derivative of I corresponds to anomalous emission due to a transition from second excited singlet. YAMAGUCHI et al. [6] have determined the dipole moments of the lower two singlet excited states of the 3,5dimethyl derivative of I from the frequency shift of the absorption and fluorescence spectra in various solvents and have pointed out that the direction of the dipole moments of both excited states is opposite to the direction of the ground-state dipole moment. Recently, magnetic circular dichroism (MCD) has been shown to be a useful tool for the assignment of absorption bands and for the investigation of the magnetic properties of the ground and excited states of molecules [7-91. The aim of this paper is to make the assignment of the absorption bands of the 3,5-dimethyl derivative of I by measuring the MCD spectra and by comparing the experimental results with the theoretical results for I obtained by use of the PARISER-PARR-POPLE
(PPP)
method [lo, 111 in combination bond-length technique [12].
hexane and by subsequent sublimation in and was spectroscopically pure, from comparison with published absorption spectra [5]. Cyclohexane was carefully purified according to the descriptions in Reference [13]. Absorption spectra were measured on a Hitachi EPS-3T recording spectrophotometer. MCD spectra were obtained with JASCO J-40 A and C recording spectropolarimeters equipped with a 15.2 kG electromagnet. All measurements were made on deaerated samples at room temperature. The Faraday B values were extracted from the MCD spectra by use of the formula: from
SCF-T-MO
mcuo
B =-(33.53)-l
[
Jtmd
([&/v)dv
where u was frequency in cm-’ and [e], was molar ellipticity per unit field in units of deg. I.m-’ mol-’ G-’ [8]. The MCD and absorption spectra were analyzed assuming G-type absorption bands with the method described by BADOZ et al. [14]. MO CALCULATION
We have calculated the electronic transition energies (AE), the dipole strengths (D), and the Faraday B values (B) of I by use of the PARISERPARR-P• PLE method [lo, 111 incorperated with the variable bond-length technique [12]. The 49
CI
with the variable
EXPERIMENTAL
The 3,5dimethyl derivative of I was prepared by one of the present authors (K. H.) [l]. This compound was carefully purified by recrystallization * To whom inquires should be directed.
Fig. 1. Numbering of carbon atoms and choice of axes. 1147
H. YAMAGUCHI,T. KOYANAGI, T. MURAOKA and K. HAFNER
1148 singly excited figuration
configurations
interactions.
electron
repulsion
evaluated
by
11.16
potential
and
(I,)
were
exponential
formula
[12].
moments
Faraday
B
employed values
the procedure evaluated
quantities ING [19],
described
and
the
respectively.
Two-
The
by use of the
magnetic
and elec-
evaluation
calculated
of the
according
to
by KAITO et al. [16]
and
operator
Liiwon+orthogonalized
atomic
[17], basis
were used for the computation of both [18]. According to CALDWELL and EYRthe
Faraday
B
terms
calculated
for
RFSULTS AND DISCUSSION The
absorption
and
MCD
spectra
of the
I
I
I
1
32
36
40
4L
a
noncentric molecule using an incomplete set of state functions were origin dependent. In this paper, origin was set at the midpoint between the centers of charges for the ground and lower five singlet excited states.
dimethyl
35
derivative
of I are shown in Figs. 2 and 3. As the MCD spectra observed in higher wavenumber region than 44.0 x lo3 cm-’ are rich of noises, we do not show the MCD spectra in Fig. 3. The calculated transition energies (AE), dipole strengths (D), and Faraday B values (B) for I are listed in Table 1 along with the experimental results for the 3,Sdimethyl derivative of I. The calculated transition energies are in fairly good agreement
[‘I, 0.0
40
KCM-’
Fig. 3. Magnetic circular dichroism spectrum (top) and absorption spectrum (bottom) of 3,5dimethylcyclopenta[efl-heptalene in the wavenumber region of 29.6-44.0x lo3 cm-tin
cyclohexane
at room temperature.
with the experimental transition energies. The agreement between the calculated and experimental values for D and B is not so good as expected. However, the predicted signs of the Faraday B terms extracted from the MCD spectra. The first absorption band exists in the wavenumber region of about 8.0~ lo3 to 20.0~ lo3 cm-‘. The MCD spectra of this band show a negative curve. The calculated transition energy for this band is 12.4~ lo3 cm-’ and the calculated Faraday B term for this transition shows the positive value (the sign of the Faraday B term is opposite to the sign of the MCD spectra). As the predicted transition energy and sign of the MCD spectra for this band are in good agreement with the experimental results for this band, the first band can be assigned to the ‘A, -+ ‘B2 transition. The MCD spectra appeared in the wavenumber region of 20.0-24.0 x lo3 cm-i show the positive and negative curves. The calculated transition energy and Faraday B term are 23.4 X lo3 cm-’ and positive, Table 1. Transition energies (AE), dipole and Faraday
12
16
20
2L
28
KCM’-’
Fig. 2. Magnetic circular dichroism spectrum (top) and absorption spectrum (bottom) of 3,5-dimethylcyclopenta[ef]-heptalene in the wavenumber region of about 8.0X lo3 to 29.6 x lo3 cm-l in cyclohexane at room temperature.
Type
AE*
lA, +lB,
12.4 23.4 28.1 30.9 36.1 36.9 40.0
IAl + ‘A, ‘Al+ ‘B, ‘A, + IA, IAl + rB, IA, -+‘A1 ‘A,+‘B,
*In lo3 cm-r. tin Debye’. Sin 1O-3 Bohr
Dt
(D)
Experimental B$
2.96 3.85 2.19 0.364 39.5 3.32 15.3 -8.21 41.3 -0.948 55.8 24.7 30.7 -22.9
magneton
strengths
B values (B)
Theoretical
Transition
0
23
- -2.0
affinas Ip =
in the
_
valence
taken
by use of a dipole velocity
respectively. functions
were
computed
were
respectively [lo]
and the electron
E, = 0.03 eV,
integrals
in con-
two-center
[ 151. The
atom
center-core tric
were
equations
for the carbon
eV
and
PARISER-PARR
NISHIMOTCA~ATAGA ity (E,)
one-
integrals
the
state ionization
were included
The
Debye’
AE*
Dt
BS
12.6 23.5 27.4 31.5 36.8 38.5 42.0
0.492 0.507 16.3 7.85 21.5 24.9 14.2
0.596 0.052 2.31 -0.639 -0.692 2.22 -2.19
cm-‘.
Magnetic circular dichroism spectra of 3$dimethylcyclopenta[ef]heptalene respectively. From these facts, it can be concluded that the band at 23.5 X lo3 cm-’ is ascribed to the second transition (IA,+ IAl). As pointed out by CALDWELL and EYRING [19], it is considered that the positive sign of the MCD spectra at 22.0X lo3 cm-’ comes from coupling with the vibrational modes different from the vibrational modes coupled at 23.5 x lo3 cm-‘. The MCD spectra have the negative curve in the wavenumber region of 24.0-29.3 x lo3 cm-l. The calculated transition energy (28.1 x lo3 cm-l) agrees with the experimental value (27.4 x lo3 cm-l). As the calculated Faraday B term shows the positive value, the absorption band presented in this wavenumber region can be assigned to the third transition (IAl -+ ‘BZ). The absorption and MCD spectra located in the wavenumber region of 29.5-36.0 x lo3 cm-’ have an abundance of vibrational structures. The MCD spectra indicate the positive peak at 3 1.5 X lo3 cm-’ and the negative curve in the wavenumber region of 32.0~ lo3 to 36.0~ lo3 cm-‘, respectively. As the calculated transition energy and Faraday B value are 30.9~ lo3 cm-’ and negative, respectively, the absorption band being at 31.5 x lo3 cm-’ can be assigned to the fourth transition (IAl + ‘A,). The appearance of the negative curve in the wavenumber region of 32.0-36.0 x lo3 cm-’ may be attributed to the coupling between the fourth transition and the vibrational modes different from the vibrational modes coupled with the fourth transition [19]. The transition energy and Faraday B value for the fifth transition are 36.1 x lo3 cm-’ and negative, respectively. As the MCD spectra take a positive peak at 36.8 x lo3 cm-‘, it can be expected that the fifth transition (‘A, + lBz) locates at 36.8X lo3 cm-‘. Since the calculated transition energy and Faraday B value are 36.9 x lo3 cm-’ and positive, respectively, the negative MCD peak appeared at 38.5 x lo3 cm-’ belongs to the sixth transition (‘A, + ‘A,). The positive MCD peak is observed at 42.0~
SA(A) 35/l&-c
1149
lo3 cm-‘. From the presence of this MCD peak, it can be considered that the seventh transition (‘A,+ ‘B2) exists in the wavenumber region of about 39.0 x lo3 to 44.0 X lo3 cm-‘. From the above assignments, it can be concluded that the absorption and MCD spectra of I are little affected by the introduction of the methyl groups at the 3 and 5 positions.
REFERENCES
PI K. HAFNER and J. SCHNEIDER,Agnew. Chem. 70, 702 (1958); Justus Liebigs Ann. Chem. 624, 37
(1959).
PI M. A. ALI and C. A. COULSON,Mol. Phys. 4, 65 (1961). [31 A. JULG and P. FRANCOIS,Theor. Chim. Acta 7, 101
(1966). [41 H. YAMAGUCHI, T. TERASAKA and T. NAKAJIMA, Theor. Chim. Acta 18, 255 (1970). [51 D. F. EATON, T. R. EVANS and P. A. LEERMARKERS, Mol. Photochem. 1, 347 (1969). [61 H. YAMAGUCHI,T. IKEDA and H. MAMETSUKA,Bull.
Chem. Sot. Japan 49, 1762 (1976). [71 A. D. BUCKINGHAMand P. J. STEPHENS,Ann. Rev.
Phys. Chem. 17, 399 (1966). PI P. N. SCHATZ and A. J. MCCAFFREY,Quart. Revs., Chem. Sot. 23, 552 (1969). [91 D. CALDWELL,J. M. THORNE and H. EYRING, Ann. Rev. Phys. Chem. 22, 259 (1971). [lOI R. PARISERand R. G. PARR, J. Chem. Phys. 21,446,
767 (1953). [ll] J. A. POPLE, Trans. Faraday Sot. 49, 1375 (1953). [12] H. YAMAGUCHI, T. NAKAJIMA and T. L. KUNII, 7’heor.Chim. Acta 12, 349 (1968). [13] J. A. RIDDICKand W. B. BUNGER, Organic Solvent, Vol. II, (edited by A. WEISSBERGER).Wiley, New York (1970). [14] J. BADOZ, M. BILLARDON,A. C. BOCCARA and B. BRIAT, Symp. Faraday Sot. 3, 27 (1969). r151 N. MATAGA and K. NISHIMOTO,Z. Phvsik. Chem. . a
13, 140 (1957). [16] A. KAITO, A. TAJIRIand M. HATANO, J. Am. Chem. sot. 97, 5059 (1975). [17] A. IMAMURA, T. HIRANO, C. NAGATA and T. TSURUTA, Bull. Chem. Sot. Japan. 45, 396 (1972). [18] P. 0. L~WDIN, J. Chem. Phys. 18, 365 (1950). [19] D. J. CALDWELLand H. EYRING, J. Chem. Phys. 58, 1149 (1973).