Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf186 Y. G. SMEYERS ET AL.
In Table 5, we give the HPHF energy values obtained for the singlet ground state, and the first triplet and singlet excited states, as well as the RHF and UHF values
obtained elsewhere [20] with different basis sets. It is seen that whereas |
con- |
|
formation is the preferred in the singlet ground state, the |
one is the most |
|
stable in the excited states in accordance with the large band progressions observed in the electronic spectrum, as well as the band assignments [20]. This change of conformational preference can be also interpreted on the basis of the destabilization of the electronic system.
Here, once more the barrier height values are seen to be very basis dependent. The value encountered for the singlet excited state, however, is found to be in relatively good agreement with the experimental value:
3.4.FORMIC ACID
Finally, the HPHF approach is applied to the formic acid molecule, in its first singlet excited state, which is not orthogonal any more to the singlet ground state in a random conformation.
The calculations were performed into two basis sets, with full geometry optimization
except for the torsional angles |
and |
Two non planar conformations were con- |
sidered, which correspond to minima on the potential energy surface into the GVB approximation [21]. In these conformations, the molecule adopts a pyramidal conformation, as in methanal. In addition, the hydroxilic group is rotated up or down the OCO plane.
In Table 6, the formation energy values for these two preferred conformations are given, together with the corresponding values for the planar conformations syn and anti. It is seen that one of the minima is only slightly more stable than the other when calucated with the larger basis, but much more stable than the planar conformations in accordance with the GVB calculations [21].
The geometrical parameters found for these four conformations are gathered in Table 7. It is seen that the carbon atom of the preferred conformations exhibits an hybridization because of the destabilization of the electronic system by the antibonding orbital, whereas the carbon atom of the planar conformations shows mostly an one.
HARTREE-FOCK MODEL TO THE LOWEST SINGLET AND TRIPLET EXCITED STATES |
187 |
Distances in and angles in degrees.
4. Discussions and conclusions
In the present paper, the Half-Projected model is applied to the direct determination
of the lowest singlet and triplet excited states in which |
just as the usual UHF |
method is employed for states in which |
As examples, the method |
is successfully applied to the calculation of some molecular properties of methylene, methanal, dimethylglyoxal and formic acid, in these excited states.
In the case of methylene, methanal or dimethylglyoxal, in which the excited wavefunction is always orthogonal by symmetry to that of the singlet ground state, the procedure converges well without any complication. In this case, the HPHF method could be regarded as an extension of the usual UHF procedure for the states in which This extension appears to be especially interesting for the direct determination of the lowest singlet excited states of medium size molecules, for which no simple and efficient method exists.
In the case of formic acid, in a random conformation, the excited wave-function possesses the same symmetry as that of the ground state. In the present paper, we propose to orthogonalize the excited orbital to its companion in order to avoid the variational collapsing of the excited state into the fundamental one. The procedure converges more slowly, but procedures for accelerating convergence may be used. Notice that the excited wave-function is not necessarily orthogonal to the fundamental one in this way of proceeding. But, both functions have not to be orthogonal because the Hamiltonian operators (24) and (37) are not the same. They may expected, however, to be nearly orthogonal. It may be added here that a complete orthogonalization will yield probably worst results [15].
It may be concluded thus that the Half-Projected Hartree-Fock model proposed more than two decades ago for introducing some correlation effects in the ground state
wave-function [1,2], could be employed advantageously for the |
direct determination |
of the lowest triplet and singlet excited states, in which |
This procedure |
could be especially suitable for the singlet excited states of medium size molecules for which no other efficient procedure exists.
References
1. Y.G. Smeyers, An. Fis. (Madrid), 67, 12 (1971).
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2.Y.G. Smeyers and L. Doreste, Int. J. Quant. Chem. , 7, 687 (1973).
3.Y.G. Smeyers and G. Delgado-Barrio, Int. J. Quant. Chem. , 8, 733 (1974).
4.P. A. Cox and M.H. Wood, Theoret. Chim. Ada , 41, 269 (1976).
5.Y.G. Smeyers and A.M. Bruceña, Int. J. Quant. Chem. , 14, 641 (1978).
6.B.H. Lengsfield III, D.H. Phillips and J.C. Schug, J. Chem. Phys. , 74, 5174 (1981).
7.S. Olivella and J. Salvador, Int. J. Quant. Chem. , 37, 713 (1990).
8.M.B. Ruiz, P. Fernández-Serra and Y.G. Smeyers, Fol. Chim. Theor. Lat., 19, 85 (1991).
9.S. Olivella and J. Salvador, J. Comp. Chem., 12, 792 (1991).
10.R.G.A. Bone and P. Pulay, Int. J. Quant. Chem. , 45, 133 (1992).
11.G. Berthier, J. Chim. Phys. , 51, 363 (1954).
12.J.A. Pople and R.K. Nesbet, J. Chem. Phys. , 22, 571 (1954).
13.J.C. Slater, in Quantum Theory of Atomic Structure, McGraw-Hill, New York, 1960.
14.A.T. Amos and G.G. Hall, Proc. Roy. Soc. , A263, 483 (1961).
15.R. Colle, A. Fortunelli and O. Salvetti, Theoret. Chim. Ada , 71, 467 (1987).
16.G. Duxbury and Ch. Jungen, Mol. Phys. ,6, 981 (1988).
17.P.R. Bunker and T.J. Sears, J. Chem. Phys. , 83, 4866 (1985).
18.M. Baba, U. Nagashima and I. Hanazaki, Chem. Phys. , 93, 425 (1985).
19.V.T. Jones and J.B. Coon, J. Mol. Spectrosc. , 31, 137 (1969).
20.M.L. Senent, D.Moule, Y.G. Smeyers, A. Toro-Labbé and F.J. Peñalver, J. Mol
Spectrosc., in press.
21.Fr. loannoni, Fluorescence Spectra and Ab Initio Study of HCOOH, Master Degree Thesis, Brock University, St-Catharines, 1989.
FSGO Hartree-Fock Instabilities of Hydrogen in External Electric Fields
J.M. ANDRE, G. HARDY, D. H. MOSLEY and L. PIELA
Facultés Universitaires Notre-Dame de la Paix, Laboratoire de Chimie Théorique
Appliquée, 61 rue de Bruxelles, B-5000 Namur, Belgium and University of Warsaw,
Quantum Chemistry Laboratory, Pasteura 1, 02-093 Warsaw, Poland
1. Introduction
In the early sixties, it was shown by Roothaan [ 1 ] and Löwdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution.
The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial
function (orbital) but differ only by its multiplication by the a or spin basis functions.
It is common knowledge that, in the case of the hydrogen molecule studied in a minimal basis set, the correlation error can be explained by the existence of ionic species in the hydrogen dissociation products:
This is an artefact due to the non-zero probability of the restricted wave-function of finding two electrons of opposite spins at the same spatial position.
FSGO's (Floating Spherical Gaussian Orbital) were introduced by Frost [3] in the mid
1960s. With FSGO's one abandons the idea of atomic orbitals centred on nuclear positions to arrive at an even more compact basis set than a minimal one. FSGO's correspond to s-type Gaussians that are not fixed at the atomic centers but are able to
"float" in space so as to optimally represent each localized pair of electrons. Because only one function is needed for each electron pair, the basis set used is often referred to as being "subminimal".
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© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
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With respect to correlation, the behaviour of the hydrogen molecule studied in a subminimal FSGO basis set is still more striking than the one observed in a minimal basis set. By symmetry arguments, the single FSGO which describes the electron pair of the hydrogen molecule is centred at the middle of the H-H bond. As the internuclear distance increases and ultimately when the molecule dissociates, such a description would lead to a physical nonsense. Indeed, at the dissociation limit, this would correspond to two
protons |
and an isolated pair of electrons |
Thus, we understand that, in the FSGO model, for some critical distance, the single Gaussian will jump from its symmetric position at the middle of the H-H bond to a dissymetric one represented below. Thus, the FSGO dissociation scheme corresponds to
one electron pair on one of the proton |
and no electron on the second proton |
This behaviour is a nice example of the symmetry dilemma in the conventional HartreeFock scheme and is intimately connected with the question of Hartree-Fock instability.
In this paper, we analyze the instabilities which appear in the Hartree-Fock method. Our analysis is made in the framework of basis sets. In the hydrogen molecule, the single floating Gaussian orbital (FSGO) desribing the electron pair has its optimal position in the middle of the hydrogen molecule only for small internuclear distances. For large enough distances its optimum position is close to one of the nuclei and a broken- symmetry solution is thus preferred. Application of an external electric field along the molecular axis induces some additional instabilities in the lowest-energy solution with respect to the electric field value. Here, both types of instabilities are investigated analytically as well as numerically.
This paper is, thus, a double tribute to Professor Berthier. On one side, G. Berthier has provided excellent analysis of quantum mechanical instabilities [4], while additionally being at the origin of the interest of the Namur group for studies of (hyper)polarizabilities in organic molecules and chains.
2. Subminimal basis set Hartree-Fock-type calculations of the hydrogen molecule
In a FSGO basis set, the Gaussian orbital is defined by:
where both, the Gaussian exponent and the Gaussian center are optimized in a FSGO calculation. Depending on the complexity of the calculation, we could have to compute the following integrals:
one-electron integrals:
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191 |
two-electron integrals:
In the RHF method, the FSGO describing the electron pairs is doubly occupied and the wave-function has the form:
Its associated RHF energy is, at a given internuclear distance (R):
Table 1. shows the total energies obtained using the RHF method for: 1. LCAO minimal basis set STO-1G for the sake of comparison with FSGO, 2. FSGO in its symmetric and broken symmetry solutions and, 3. LCAO minimal basis set STO-3G in order to allow a safer comparison with the quality of the subminimal basis used in the FSGO technique. The dissociation curves are given in Figure 1.
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The “equilibrium” FSGO-hydrogen molecule is found for the parameters:
(could be compared to the experiment: 1.401 a.u.) (center of the molecule)
One observes that the energy minimum at the calculated equilibrium internuclear distance always corresponds to the symmetric solutions For the values given between parentheses in the above table, the broken symmetry solution does not exist; the single Gaussian remains centered at the middle of the H-H bond. However, for interatomic distances greater than 5.6 a.u., the broken symmetry solutions (Gaussians centered near
give the absolute minimum while the symmetric solution has a higher energy; this is a further example that, for approximate wave functions, the basic symmetry properties do not follow automatically from the variation principle and consequently do not have necessarily the full symmetry of the nuclear framework.
In the HF scheme, the first origin of the correlation between electrons of antiparallel spins comes from the restriction that they are forced to occupy the same orbital (RHF scheme) and thus some of the same location in space. A simple way of taking into account the basic effects of the electronic correlation is to release the constraint of double occupation and so use Different Orbitals for Different Spins (DODS scheme which is the European way of calling UHF). In this methodology,
electrons with antiparallel spins are not found to doubly occupy the same orbital so that, in principle, they are not forced to coexist in the same spatial region as is the case in usual RHFwavefunctions.
A UHF wave function over different orbitals |
and |
is then: |
The wave function obtained corresponds to the Unrestricted Hartree-Fock scheme and
becomes equivalent to the RHF case if the orbitals and are the same. In this UHF form, the UHF wave function obeys the Pauli principle but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-electron case, an alternative form of the wave function which has the same total energy, which is a pure singlet state, but which is no longer antisymmetric as required by thePauli principle, is:
In both cases, the energy formula (E(UHF-FSGO)) is the same:
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193 |
Table 2 gives the values of E(RHF-FSGO) and E(UHF-FSGO) for internuclear distances
from 1.0 a.u. to 7 a.u. (step 0.5 a.u.) and also the value for |
We observe that we |
have the "correct" dissociation behavior for the UHF case
Since in those forms of the UHF wave functions, one drops a constraint (either the need of a pure spin state in the first case or the Pauli antisymmetry rule in the second case), it is expected that the resulting wave function will give a lower energy than in the RHF case and thus introduce a part of the correlation energy. As shown in the table above, there is
no splitting of the orbitals for small interatomic distances The single Gaussian describing the different spinorbitals remains located at the same central
position. For larger distances, however, a progressive splitting of the orbitals exists, with the orbitals tending to localize near each hydrogen atom leading to correct dissociation into two hydrogen atoms.
Further improvements to the previous UHF schemes can be obtained by using the Projected (PHF) and Extended (EHF) Hartree-Fock schemes. Löwdin has shown that if one carries out a component analysis of the non-pure UHF wave function, there is at least
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one component which restores the symmetry and which has a lower energy than the UHF wave function.
The form of the PHF wave function is, in this case:
Note that the PHF wave function is no longer a single determinant and is a sum of two terms. This PHF function both satisfies the Pauli principle and is a pure singlet state. The energy formula E(PHF-FSGO) is easily derived:
In the PHF method, the variational procedure is applied to the UHF wave function and
subsequently the projection is performed on the UHF |
and |
orbitals. If no splitting is |
|
obtained during the UHF step |
the RHF, UHF and PHF wave functions are |
||
equivalent and have the same total energy.
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195 |
In the Extended Hartree-Fock (EHF) technique, the minimization is performed on the form of the PHF wave function. This type of wave function should produce for each interatomic distance a further lowering of the energy with respect to the RHF, UHF, and
PHF total energies. The values of E(PHF-FSGO), and E(EHF-FSGO) for internuclear distances from 1.0 a.u. to 7 a.u. (step 0.5 a.u.) are also given in Table 2. As in the UHF
case, we have the "correct" dissociation behavior
Indeed, the extrapolated values converge to the expected values; we have shown that in
the RHF symmetric case, the limiting situation is |
an asymptotic energy |
which should correspond to the H- described whose two electrons are described by a single Gaussian:
With the optimisation condition: