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The Ab initio spin-coupled description of the process HCN→CH+N
Journal of Molecular Structure (Theochem), 229 (1991) 279-289 Elsevier Science Publishers B.V., Amsterdam
THE AR INITIO SPIN-COUPLED PROCESS HCN+CH+N
DESCRIPTION
279
OF THE
MAURIZIO SIRONI and MARIO RAIMONDI Dipartimento di Chimica Fisica ed Elettrochimica, UniversiM di Milano, Via Golgi 19,20133 Milan0 (Italy) DAVID L. COOPER Department of Chemistry, University of Liverpool, P.O. Box 147, Liverpool L69 3BX (U.K.) JOSEPH GERRATT Department of Theoretical Chemistry, University of Bristol, Cantocks Close, Bristol BS8 1 TS (U.K.) (Received 12 September 1990)
ABSTRACT The process of breaking the strong C-N triple bond in HCN was investigated using spincoupled theory. Although it is based on just a single spatial configuration, the spin-coupled wavefunction describes correctly the dissociation into CH (a4.Z-) and N (4S) fragments as the N atom is removed. The changes occurring at a molecular level are manifested by modifications both to the form of the orbit& and to the mode of spin coupling. The resulting spin-coupled orbitals resemble most closely the functions postulated by classical valence bond theory. However, near the equilibrium geometry of HCN, two of the spin-coupled orbitals have the character of twocentre molecular orbitals localized in the CN moiety.
INTRODUCTION
The HCN molecule is characterized by a strong triple bond between the carbon and nitrogen atoms and the process of breaking this bond, HCN+CH (4Z-) + N (4S), represents a challenge for any theory of valence. The simplest molecular orbital (MO) theory description, the restricted Hartree-Fock self-consistent-field (SCF) wavefunction, predicts incorrect dissociation products, namely ions. Uniform descriptions within MO theory as the C-N separation is increased require multiconfiguration wavefunctions. These inevitably present problems of chemical interpretation. Ab initio implementations of classical valence bond (VB) theory might appear to provide a suitable alternative, but the physical picture tends to be obscured by the intervention of significant contributions from large numbers of ionic structures. In this paper we present a study of the process HCN+CH ( 4.Z-) + N ( 4S)
0166-1280/91/$03.50
0 1991-
Elsevier Science Publishers B.V.
280
using spin-coupled theory, which represents the modern development of VB theory. The spin-coupled approach to electronic structure has now been applied to a very wide range of molecular problems and several recent reviews are available [ 11. The spin-coupled wavefunction is based on a single spatial configuration of singly-occupied non-orthogonal orbitals and uses the complete spin space for the ten valence electrons. The changes occurring at a molecular level are manifested by modifications both to the form of the orbitals and to the mode of coupling of the electron spins. It is evident that the demands posed are far from trivial. First of all, the orbitals appropriate for HCN at its equilibrium geometry must undergo significant changes so as to be transformed into the pure atomic orbitals of the 4S ground state of N and into the spin-coupled orbitals of CH in its a4L’excited state. Secondly, the dissociation of the molecular singlet state into two quartet fragments necessarily involves extensive recoupling of the electron spins. The MO theory wavefunction of comparable accuracy to the spin-coupled description presented here is the CASSCF wavefunction with all distributions of the ten valence electrons in an active space of ten orbitals. Carrying out such a calculation presents no particular problems, provided efficient modern algorithms are employed [2], but this CASSCF wavefunction consists of more than 12 000 configurations and so the problem of extracting a useful physical and chemical picture is non-trivial. CALCULATIONS
Starting from linear HCN at its equilibrium geometry, with r (C-H) = 2.025 bohr and r( C-N) = 2.1756 bohr, we first carried out a restricted Hartree-Fock SCF calculation. The spherical gaussian basis set consisted of (lls6p/5s) primitive functions contracted to [ 5s3p/3s] and augmented with polarization functions with exponents dc=0.72,dN-0.98andPn=l.O [3]. The first two MOs, lo and 20, were used to provide a description of the four core electrons, essentially N ( 1s”) and C (1s 2). Spin-coupled theory was then applied explicitly for the ten valence electrons, which were described by ten distinct non-orthogonal orbitals. We retained the o-n separation resulting from the SCF calculations. Six of the spincoupled orbitals ( rrl-oe) were expanded in the basis comprising all the MOs of o symmetry, but excluding lo and 20 which describe the core. The remaining four spin-coupled orbitals, 7rl-7c4,were expanded in the basis comprising all the MOs of n symmetry. No further constraints were imposed on the orbitals. The spin-coupled wavefunction used here can be written in the form
k=l
281
where the index k labels the particular mode of spin coupling, of which there are 42 for a ten-electron system with a total spin of zero. I!$&is the perfectpairing spin function for four electrons with a net spin of zero. The cO,k,which may be termed spin-coupling coefficients,, were optimized simultaneously with the ten spin-coupled orbitals 0,-o, and 7cl-x4. In principle, many different bases of spin functions @P,;, could be used, with the most appropriate choice depending on the process under study. For an account of different spin bases, and of the relationships between them see, for example, ref. 4. For HCN near its equilibrium geometry, it is useful to employ the Rumer basis which emphasizes the pairing of electrons to form bonds. The version of the program used for the present calculations [ 51 utilizes the Rumer basis. However, as r (C-N) becomes large, it is more appropriate to transform the spin-coupling coefficients to a “non-standard genealogical” (NSG) basis in which the spins of the separated fragments appear. We employ the full spin space of 42 functions for all geometries. RESULTS
We report in Table 1 the SCF and spin-coupled energies of the HCN system for the different values of r (C-N) considered . The computed dissociation energy is 7.71 eV, which is to be compared with the value of 10.41 eV obtained by adding 0.72 eV, the X217-a4L’- separation in CH, to 9.69 eV, the experimental value for the dissociation energy of HCN into the ground-state products, CH (X’n) and N ( 4S). Although it is based on just a single spatial configuration, the spin-coupled wavefunction provides 74% of the dissociation energy. This value could be improved by including additional spatial configurations in a non-orthogonal CI calculation to take better account of the relevant dynamical correlation effects. All our experience to date suggests that this further quantitative reTABLE 1 Calculated energies for HCN (C,,) r(C-N)
2.1756 2.5 3.0 3.5 4.0 5.0 10.0
(bohr)
with r( C-H) = 2.025 bohr Energy (hartree ) SCF
Spin-coupled
- 92.2658 -92.8327 -92.6512 -92.5012 -92.4226 - 92.3726 - 92.2659
-92.7134 -92.9686 - 92.8574 - 92.7785 -92.7400 -92.7174 -92.7134
Fig. 1. Spin-coupled orbitals ‘TV,02, o,, 04, o,, a,, 7~~and n, for HCN at its equilibrium geometry. :n, and x4 are related by symmetry to z, and x2. In each frame, the molecule points up the page with the atoms in the order HCN.
‘3
0
0
_
.
I I rl
G
L
I
d
Fig. 3. Spin-coupled orbitals for HCN with r(C-N) order HCN.
increased to 3.5 bohr. In each frame, the molecule points up the page with the atoms in the
Fig. 5. Spin-coupled orbitals for HCN with r(C-N) increased to 10.0 bohr. The molecule is oriented as in Figs. l-4. The frames depicting 4, 02, 0, and 7~~ show only the vicinity of the N fragment. The frames depicting r~3, G5, c6 and n, show only the vicinity of the CH fragment.
287 TABLE2 Selected overlap integrals as a function of r( C-N). Near equilibrium geometry, 4, a, are nonbonding electrons on N; o,, a, are the two-centre orbitals which describe the C-N u bond; n,, n, describe one of the C-N n bonds
2.1756 2.5 3.0 3.5 4.0 5.0 10.0
0.76 0.81 0.88 0.90 0.96 0.97 0.97
0.80 0.71 0.67 0.58 0.40 0.18 0.00
0.68 0.58 0.41 0.26 0.16 0.06 0.00
finement would not significantly alter the qualitative picture described below. A similar statement applies to the inclusion of ionic structures. The spin-coupled orbitals or-o, and 7rl-7r2for HCN at its equilibrium geometry are shown in Fig. 1; orbitals 7r3and 7c4have not been shown because they are trivially related by symmetry to xl and ;n,. Orbitals ol and o, are sp”like hybrids localized on nitrogen and point away from the CH unit: each of these hybrids accommodates a nonbonding electron. Orbitals 0, and o,, which have an overlap with one another of 0.80, have the character of two-centre molecular orbitals localized on the CN moiety. Orbitals 0, and 06 describe the C-H bond and have an overlap of 0.80: 0, is an sp”-like orbital localized on C, but is distorted towards the H atom, and 06 is a Is(H) function polarized by the C atom. Orbitals x1 and 7r2have an overlap of 0.68 and take the form of distorted 2p, functions on C and N, respectively. Orbitals 7r3and 7r4are the corresponding orbitals based on 2p, functions. Spin-coupled orbitals resulting from the calculations with different values of r (C-N) are shown in Figs. 2-5. As anticipated above, there are major changes in the form of the orbitals as the C-N separation is increased. The nonbonding orbitals on N, o, and o,, become essentially N (2s) functions; 0, evolves into a nonbonding sp”-like hybrid localized on carbon and 0, into a 2p, function localized on nitrogen; x1 and 7c2become pure 2p, functions on C and N, with equivalent changes in 7r3and 7r4.As might be expected on the basis of chemical intuition, orbitals 0, and 06, which describe the C-H bond, change very little during the process of removing the N atom. Selected overlap integrals are collected in Table 2 and show the expected persistence of the C-N o bond ( a3/ 04) in comparison with the n bonds ( 7r1/7r2). We use the symbol 1CH (4Z-);N (4S) ) to denote the spin function in the NSG basis which represents the interaction of the two quartets localized on CH and N. We show in Fig. 6 the variation in the weight of this spin function with r (C-N). The asymptotic situation is perfectly described by
288 12 weight
1.0
0.8
0.6
IH-EN:>
6.0
8.0
la0 r(C-N)/bohr
Fig. 6. Weights of the two most important modes of spin coupling as a function of r (C-N). further details see the text.
For
1CH (4X- ) ; N ( 4S) ) , which then has a weight of unity. Superimposed on Fig. 6 is the variation with r(@-N) in the weight of the Rumer spin function that corresponds to perfect-pairing of the electron spins, with the orbitals in the order o 12345612 o rs rr o o 7~7~7~ : ) to denote this spin 3 n4. We use the symbol 1H-&N function. Even at equilibrium geometry, 1H-&N : ) only accounts for 65% of the spin-coupled wavefunction, with significant contributions from other modes of spin coupling. CONCLUSIONS
We find that the spin-coupled description presented here for the valence electrons in HCN coincides fairly closely with the conventional intuitive chemical view of localized covalent bonds and of nonbonding electrons. Hybridization, leading to the formation of @-like orbitals on C and N, arises naturally, simply by minimizing the total energy without preconceptions on our part as to the degree of localization or delocalization of the orbitals. There are, however, some important differences from the classical VB description of this molecule. First of all, the two orbitals involved in the C-N o bond (0, and cr4)turn out to be two-centre molecular orbitals localized in the CN moiety. Secondly, all the hybrids on a given centre turn out to be nonorthogonal. Thirdly, all the spin-coupled orbitals show distortions towards neighbouring atoms. This has the important conceptual consequence that there
289
would only be minor changes in the wavefunction if ionic structures were introduced. Finally, a notable feature of our description of the HCN molecule near its equilibrium geometry is that the Rumer spin function corresponding to the Lewis structure H-&N : accounts for only 65% of the total wavefunction. This suggests that a quantitative treatment of multiple bonds may require a careful treatment of electron correlation effects and, in particular, that models which do not utilize the full spin space are likely to produce misleading orbital pictures. Despite being based on just a single spatial configuration, the spin-coupled wavefunction describes correctly the process HCN-t CHCN and provides a simple physical picture of the changes that occur as the N atom is removed.
REFERENCES 1
2 3 4 5
(a) D.L. Cooper, J. Gerratt and M. Raimondi, Adv. Chem. Phys., 69 (1987) 319. (b) D.L. Cooper, J. Gerratt and M. Raimondi, Int. Rev. Phys. Chem., 7 (1988) 59. (c) J. Gerratt, D.L. Cooper and M. Raimondi, in D.J. Klein and N. Trinajstic (Eds.), Valence Bond Theory and Chemical Structure, Elsevier, Amsterdam, 1990. (d) D.L. Cooper, J. Gerratt and M. Raimondi, in I. Gutman and S.J. Cyvin (Eds.), Advances in the Theory of Benzenoid Hydrocarbons, Top. Curr. Chem., 153 (1990) 41. (e) D.L. Cooper, J. Gerratt and M. Raimondi, Mol. Simul., 4 (1990) 293. (a) H.-J. Werner and P.J. Knowles, J. Chem. Phys., 82 (1985) 5053. (b) P.J. Knowles and H.-J. Werner, Chem. Phys. Lett., 115 (1985) 259. T.H. Dunning, J. Chem. Phys., 55 (1971) 716. R. Pauncz, Spin Eigenfunctions, Plenum, New York, 1979. (a) M. Sironi, Tesi di Dottorata di Ricerca, University of Milan, 1989. (b) M. Sironi, D.L. Cooper, J. Gerratt and M. Raimondi, to be published.
THE AR INITIO SPIN-COUPLED PROCESS HCN+CH+N
DESCRIPTION
279
OF THE
MAURIZIO SIRONI and MARIO RAIMONDI Dipartimento di Chimica Fisica ed Elettrochimica, UniversiM di Milano, Via Golgi 19,20133 Milan0 (Italy) DAVID L. COOPER Department of Chemistry, University of Liverpool, P.O. Box 147, Liverpool L69 3BX (U.K.) JOSEPH GERRATT Department of Theoretical Chemistry, University of Bristol, Cantocks Close, Bristol BS8 1 TS (U.K.) (Received 12 September 1990)
ABSTRACT The process of breaking the strong C-N triple bond in HCN was investigated using spincoupled theory. Although it is based on just a single spatial configuration, the spin-coupled wavefunction describes correctly the dissociation into CH (a4.Z-) and N (4S) fragments as the N atom is removed. The changes occurring at a molecular level are manifested by modifications both to the form of the orbit& and to the mode of spin coupling. The resulting spin-coupled orbitals resemble most closely the functions postulated by classical valence bond theory. However, near the equilibrium geometry of HCN, two of the spin-coupled orbitals have the character of twocentre molecular orbitals localized in the CN moiety.
INTRODUCTION
The HCN molecule is characterized by a strong triple bond between the carbon and nitrogen atoms and the process of breaking this bond, HCN+CH (4Z-) + N (4S), represents a challenge for any theory of valence. The simplest molecular orbital (MO) theory description, the restricted Hartree-Fock self-consistent-field (SCF) wavefunction, predicts incorrect dissociation products, namely ions. Uniform descriptions within MO theory as the C-N separation is increased require multiconfiguration wavefunctions. These inevitably present problems of chemical interpretation. Ab initio implementations of classical valence bond (VB) theory might appear to provide a suitable alternative, but the physical picture tends to be obscured by the intervention of significant contributions from large numbers of ionic structures. In this paper we present a study of the process HCN+CH ( 4.Z-) + N ( 4S)
0166-1280/91/$03.50
0 1991-
Elsevier Science Publishers B.V.
280
using spin-coupled theory, which represents the modern development of VB theory. The spin-coupled approach to electronic structure has now been applied to a very wide range of molecular problems and several recent reviews are available [ 11. The spin-coupled wavefunction is based on a single spatial configuration of singly-occupied non-orthogonal orbitals and uses the complete spin space for the ten valence electrons. The changes occurring at a molecular level are manifested by modifications both to the form of the orbitals and to the mode of coupling of the electron spins. It is evident that the demands posed are far from trivial. First of all, the orbitals appropriate for HCN at its equilibrium geometry must undergo significant changes so as to be transformed into the pure atomic orbitals of the 4S ground state of N and into the spin-coupled orbitals of CH in its a4L’excited state. Secondly, the dissociation of the molecular singlet state into two quartet fragments necessarily involves extensive recoupling of the electron spins. The MO theory wavefunction of comparable accuracy to the spin-coupled description presented here is the CASSCF wavefunction with all distributions of the ten valence electrons in an active space of ten orbitals. Carrying out such a calculation presents no particular problems, provided efficient modern algorithms are employed [2], but this CASSCF wavefunction consists of more than 12 000 configurations and so the problem of extracting a useful physical and chemical picture is non-trivial. CALCULATIONS
Starting from linear HCN at its equilibrium geometry, with r (C-H) = 2.025 bohr and r( C-N) = 2.1756 bohr, we first carried out a restricted Hartree-Fock SCF calculation. The spherical gaussian basis set consisted of (lls6p/5s) primitive functions contracted to [ 5s3p/3s] and augmented with polarization functions with exponents dc=0.72,dN-0.98andPn=l.O [3]. The first two MOs, lo and 20, were used to provide a description of the four core electrons, essentially N ( 1s”) and C (1s 2). Spin-coupled theory was then applied explicitly for the ten valence electrons, which were described by ten distinct non-orthogonal orbitals. We retained the o-n separation resulting from the SCF calculations. Six of the spincoupled orbitals ( rrl-oe) were expanded in the basis comprising all the MOs of o symmetry, but excluding lo and 20 which describe the core. The remaining four spin-coupled orbitals, 7rl-7c4,were expanded in the basis comprising all the MOs of n symmetry. No further constraints were imposed on the orbitals. The spin-coupled wavefunction used here can be written in the form
k=l
281
where the index k labels the particular mode of spin coupling, of which there are 42 for a ten-electron system with a total spin of zero. I!$&is the perfectpairing spin function for four electrons with a net spin of zero. The cO,k,which may be termed spin-coupling coefficients,, were optimized simultaneously with the ten spin-coupled orbitals 0,-o, and 7cl-x4. In principle, many different bases of spin functions @P,;, could be used, with the most appropriate choice depending on the process under study. For an account of different spin bases, and of the relationships between them see, for example, ref. 4. For HCN near its equilibrium geometry, it is useful to employ the Rumer basis which emphasizes the pairing of electrons to form bonds. The version of the program used for the present calculations [ 51 utilizes the Rumer basis. However, as r (C-N) becomes large, it is more appropriate to transform the spin-coupling coefficients to a “non-standard genealogical” (NSG) basis in which the spins of the separated fragments appear. We employ the full spin space of 42 functions for all geometries. RESULTS
We report in Table 1 the SCF and spin-coupled energies of the HCN system for the different values of r (C-N) considered . The computed dissociation energy is 7.71 eV, which is to be compared with the value of 10.41 eV obtained by adding 0.72 eV, the X217-a4L’- separation in CH, to 9.69 eV, the experimental value for the dissociation energy of HCN into the ground-state products, CH (X’n) and N ( 4S). Although it is based on just a single spatial configuration, the spin-coupled wavefunction provides 74% of the dissociation energy. This value could be improved by including additional spatial configurations in a non-orthogonal CI calculation to take better account of the relevant dynamical correlation effects. All our experience to date suggests that this further quantitative reTABLE 1 Calculated energies for HCN (C,,) r(C-N)
2.1756 2.5 3.0 3.5 4.0 5.0 10.0
(bohr)
with r( C-H) = 2.025 bohr Energy (hartree ) SCF
Spin-coupled
- 92.2658 -92.8327 -92.6512 -92.5012 -92.4226 - 92.3726 - 92.2659
-92.7134 -92.9686 - 92.8574 - 92.7785 -92.7400 -92.7174 -92.7134
Fig. 1. Spin-coupled orbitals ‘TV,02, o,, 04, o,, a,, 7~~and n, for HCN at its equilibrium geometry. :n, and x4 are related by symmetry to z, and x2. In each frame, the molecule points up the page with the atoms in the order HCN.
‘3
0
0
_
.
I I rl
G
L
I
d
Fig. 3. Spin-coupled orbitals for HCN with r(C-N) order HCN.
increased to 3.5 bohr. In each frame, the molecule points up the page with the atoms in the
Fig. 5. Spin-coupled orbitals for HCN with r(C-N) increased to 10.0 bohr. The molecule is oriented as in Figs. l-4. The frames depicting 4, 02, 0, and 7~~ show only the vicinity of the N fragment. The frames depicting r~3, G5, c6 and n, show only the vicinity of the CH fragment.
287 TABLE2 Selected overlap integrals as a function of r( C-N). Near equilibrium geometry, 4, a, are nonbonding electrons on N; o,, a, are the two-centre orbitals which describe the C-N u bond; n,, n, describe one of the C-N n bonds
2.1756 2.5 3.0 3.5 4.0 5.0 10.0
0.76 0.81 0.88 0.90 0.96 0.97 0.97
0.80 0.71 0.67 0.58 0.40 0.18 0.00
0.68 0.58 0.41 0.26 0.16 0.06 0.00
finement would not significantly alter the qualitative picture described below. A similar statement applies to the inclusion of ionic structures. The spin-coupled orbitals or-o, and 7rl-7r2for HCN at its equilibrium geometry are shown in Fig. 1; orbitals 7r3and 7c4have not been shown because they are trivially related by symmetry to xl and ;n,. Orbitals ol and o, are sp”like hybrids localized on nitrogen and point away from the CH unit: each of these hybrids accommodates a nonbonding electron. Orbitals 0, and o,, which have an overlap with one another of 0.80, have the character of two-centre molecular orbitals localized on the CN moiety. Orbitals 0, and 06 describe the C-H bond and have an overlap of 0.80: 0, is an sp”-like orbital localized on C, but is distorted towards the H atom, and 06 is a Is(H) function polarized by the C atom. Orbitals x1 and 7r2have an overlap of 0.68 and take the form of distorted 2p, functions on C and N, respectively. Orbitals 7r3and 7r4are the corresponding orbitals based on 2p, functions. Spin-coupled orbitals resulting from the calculations with different values of r (C-N) are shown in Figs. 2-5. As anticipated above, there are major changes in the form of the orbitals as the C-N separation is increased. The nonbonding orbitals on N, o, and o,, become essentially N (2s) functions; 0, evolves into a nonbonding sp”-like hybrid localized on carbon and 0, into a 2p, function localized on nitrogen; x1 and 7c2become pure 2p, functions on C and N, with equivalent changes in 7r3and 7r4.As might be expected on the basis of chemical intuition, orbitals 0, and 06, which describe the C-H bond, change very little during the process of removing the N atom. Selected overlap integrals are collected in Table 2 and show the expected persistence of the C-N o bond ( a3/ 04) in comparison with the n bonds ( 7r1/7r2). We use the symbol 1CH (4Z-);N (4S) ) to denote the spin function in the NSG basis which represents the interaction of the two quartets localized on CH and N. We show in Fig. 6 the variation in the weight of this spin function with r (C-N). The asymptotic situation is perfectly described by
288 12 weight
1.0
0.8
0.6
IH-EN:>
6.0
8.0
la0 r(C-N)/bohr
Fig. 6. Weights of the two most important modes of spin coupling as a function of r (C-N). further details see the text.
For
1CH (4X- ) ; N ( 4S) ) , which then has a weight of unity. Superimposed on Fig. 6 is the variation with r(@-N) in the weight of the Rumer spin function that corresponds to perfect-pairing of the electron spins, with the orbitals in the order o 12345612 o rs rr o o 7~7~7~ : ) to denote this spin 3 n4. We use the symbol 1H-&N function. Even at equilibrium geometry, 1H-&N : ) only accounts for 65% of the spin-coupled wavefunction, with significant contributions from other modes of spin coupling. CONCLUSIONS
We find that the spin-coupled description presented here for the valence electrons in HCN coincides fairly closely with the conventional intuitive chemical view of localized covalent bonds and of nonbonding electrons. Hybridization, leading to the formation of @-like orbitals on C and N, arises naturally, simply by minimizing the total energy without preconceptions on our part as to the degree of localization or delocalization of the orbitals. There are, however, some important differences from the classical VB description of this molecule. First of all, the two orbitals involved in the C-N o bond (0, and cr4)turn out to be two-centre molecular orbitals localized in the CN moiety. Secondly, all the hybrids on a given centre turn out to be nonorthogonal. Thirdly, all the spin-coupled orbitals show distortions towards neighbouring atoms. This has the important conceptual consequence that there
289
would only be minor changes in the wavefunction if ionic structures were introduced. Finally, a notable feature of our description of the HCN molecule near its equilibrium geometry is that the Rumer spin function corresponding to the Lewis structure H-&N : accounts for only 65% of the total wavefunction. This suggests that a quantitative treatment of multiple bonds may require a careful treatment of electron correlation effects and, in particular, that models which do not utilize the full spin space are likely to produce misleading orbital pictures. Despite being based on just a single spatial configuration, the spin-coupled wavefunction describes correctly the process HCN-t CHCN and provides a simple physical picture of the changes that occur as the N atom is removed.
REFERENCES 1
2 3 4 5
(a) D.L. Cooper, J. Gerratt and M. Raimondi, Adv. Chem. Phys., 69 (1987) 319. (b) D.L. Cooper, J. Gerratt and M. Raimondi, Int. Rev. Phys. Chem., 7 (1988) 59. (c) J. Gerratt, D.L. Cooper and M. Raimondi, in D.J. Klein and N. Trinajstic (Eds.), Valence Bond Theory and Chemical Structure, Elsevier, Amsterdam, 1990. (d) D.L. Cooper, J. Gerratt and M. Raimondi, in I. Gutman and S.J. Cyvin (Eds.), Advances in the Theory of Benzenoid Hydrocarbons, Top. Curr. Chem., 153 (1990) 41. (e) D.L. Cooper, J. Gerratt and M. Raimondi, Mol. Simul., 4 (1990) 293. (a) H.-J. Werner and P.J. Knowles, J. Chem. Phys., 82 (1985) 5053. (b) P.J. Knowles and H.-J. Werner, Chem. Phys. Lett., 115 (1985) 259. T.H. Dunning, J. Chem. Phys., 55 (1971) 716. R. Pauncz, Spin Eigenfunctions, Plenum, New York, 1979. (a) M. Sironi, Tesi di Dottorata di Ricerca, University of Milan, 1989. (b) M. Sironi, D.L. Cooper, J. Gerratt and M. Raimondi, to be published.