Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf246 |
R. CARBÓ AND E. BESALÚ |
6. Conclusions
A mathematical device, the NSS, which can be related to Artificial Intelligence techniques, has been defined and applied in order to solve or reformulate some quantum chemical problems. This symbol is related to computer formulae generation. It has been shown that by means of the use of NSS’s many applications of such symbols can be found in mathematics as well as in Mathematical Chemistry in particular.
Apart of being able to simplify typographical structures, the NSS symbols constitute the basic elements of a completely general framework, allowing to write mathematical formulae, in such a manner that immediate translation to any high level programming language is feasible, producing a complete general code, which can be kept sequential or parallelized in a simple manner.
Pedagogical and in many cases mnemotechnical formula structures appear to be also deduced at a very generic level as a consequence of the use of this kind of devices.
The obtained mathematical patterns seems to be also fairly well adapted to Artificial Intelligence formula writing programming philosophy.
An assorted set of purely mathematical and Quantum Chemical application examples prove the generalization power and flexibility of this presently described symbolic framework.
When NSS’s together with LKD’s are adopted as working tools, both structures appear to trigger some sort of thinking machine, in such a way that once a given problem is solved, new study areas immediately appear to be a promising future application field in the focus of the imagination eye.
One can conclude that a robust and powerful theoretical machinery has been described, possessing general, far reaching imaginative possibilities.
Perhaps there are hidden in the symbolic limbo other possible similar tools, even better than these described here. We are confident in that this paper will stimulate the research interest in this direction.
7. Acknowledgments
This work is a contribution of the "Grup de Química Quàntica de l’Institut d’Estudis Catalans" and it has been financed by the "Comissió Interdepartamental per a la Recerca i Innovació Tecnològica" of the "Generalitat de Catalunya" through a grant: #QFN91-4206. E. Besalú benefits of a grant of the "Departament d’Ensenyament de la Generalitat de Catalunya".
APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY |
247 |
8.References
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14.K.Jankowski, "Electron Correlation in Atoms", pp 1-116 in: S.Wilson (Editor),
Methods in Computational Chemistry. Electron Correlation in Atoms and Molecules, Vol. 1, Plenum Press, New York, 1987.
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J'espère donner par là une preuve de ce qu'ont avancé des chimistes très distingués, qu'on n'est peut-être pas
éloigné de l'époque à laquelle on pourra soumettre au calcul la plupart des phénomènes chimiques.
J. L. Gay-Lussac
Mémoires de Physique et de Chimie,
de la Société d'Arcueil, 31 Décembre 1808
Applications to Physical Phenomena
Some 180 years after ....
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Vibrational Modulation Effects on EPR Spectra
V. BARONE, A. GRAND, C. MINICHINO and R. SUBRA
Dipartimento di Chimica, Universita Federico II,Via Mezzocannone 4, 80134 Napoli, Italy SESAM, CEN Grenoble, BP 85X, F-38041 Grenoble, France
1. Introduction
Hyperfine coupling constants provide a direct experimental measure of the distribution of unpaired spin density in paramagnetic molecules and can serve as a critical benchmark for electronic wave functions [1,2]. Conversely, given an accurate theoretical model, one can obtain considerable information on the equilibrium structure of a free radical from the computed hyperfine coupling constants and from their dependence on temperature. In this scenario, proper account of vibrational modulation effects is not less important than the use of a high quality electronic wave function.
Semirigid molecules can be described in terms of normal modes by well known perturbative treatments [3]. This approach is, however, ill-adapted to treat large amplitude vibrations, in view of their strong curvilinear character and of poor convergency in the Taylor expansion of the potential [4]. These situations demand, especially in the case of lareg (i.e. containing more than four atoms) molecules, some separation between the active large amplitude motions (LAM) and the "spectator" small amplitude ones. On these grounds, the influence of vibrational effects on EPR parameters has been studied at the abinitio level for a series of radicals [5-14], using different basis sets, correlation expansions, and treatments of vibrational averaging. In our opinion the key limitation of these approaches is their lack of generality. In fact, the use of global internal coordinates and of analytical kinetic energies leads to quite complicated formalisms specific to a reduced class of systems [12-15], unless oversimplified metrics are used [11,13,14]. We have recently proposed a general numerical procedure [16] to treat the nuclear motion taking into the proper account the variation of the reduced mass along any kind of curvilinear LAM. Here we apply this approach to the radicals CH3 and CF3, whose inversion motion is governed by quite different potential wells. In order to focus attention on general trends, avoiding specific technical details, we have used a standard polarised basis set (6-311G**) and treatment of correlation (MP2). The more so as for localized pseudo radicals, this level of theory appears completely adequate and readily applicable to large systems [17],
2. Methods
All the electronic calculations were performed with the GAUSSIAN/90 [18] and GAUSSIAN/92 [19] codes and the vibrational studies by the DiNa package [16]. Electronic wave functions were generated by the Unrestricted Hartree-Fock (UHF) formalism,
251
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 251–260.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
252 |
V. BARONE ET AL. |
correlation energy being then introduced by second order many-body perturbation (UMP2) theory [20]. All electrons were always correlated, for we have shown [21] that core electrons play an important role in the calculation of hyperfine coupling constants. The most serious criticism to this approach would be that the wave function consisting of UHF orbitals does not represent a correct spin state of the molecular system under consideration. Since, however, all the computations reported in this study give a very low spin
contamination |
we can expect quite accurate values of spin dependent |
properties. |
|
Basis set effects were not in the ground of this study, so that the 6-311G** [22] basis set has been chosen as a compromise between reliability and computation times.
Isotropic Hyperfine coupling constants |
are related to the spin densities |
at the |
corresponding nuclei by |
|
|
where |
' is the ratio of the isotropic g value for the radical to that of the free electron, |
and |
are the nuclear magnetogyric ratio and nuclear magneton, respectively. In turn, the |
spin density at nucleus N can be calculated as the expectation value of the spin density operator over the electronic wave function
where the index v runs on all electrons, and Sz is the quantum number of the total electron spin (1/2 for radicals).
In the framework of the Born-Oppenheimer approximation, we can speak of a potential energy surface (PES) and of a "property surface", which can be obtained from electronic wave functions at different nuclear configurations. In this scheme, expectation values of observables (e.g. hyperfine coupling constants) are obtained by averaging the "property surface" on the nuclear wave functions. To proceed further, let us introduce a curvilinear path continuously describing the large amplitude motion (LAM) joining two (possibly equivalent) energy minima through a first order saddle point (SP). Next, the path is parametrized in terms of the signed arc length s in mass weighted (MW) cartesian coordinates. The only necessary condition on the path is that it must not contain any translational or rotational component. For the remaining f-1 internal degrees of freedom, {Qi} (which will be referred to as the small amplitude, SA,coordinates) the potential energy contributions are approximated to second order terms along the LA path. These local vibrational coordinates must be orthogonal to the path tangent, to translations and to infinitesimal rotations. In the adiabatic approach [23], the components of the SA coordinates in the space of the MW cartesian coordinates are the eigenvectors of the Hessian matrix from which translations, rotations and path tangent are projected out. The
{Qi} are further assumed to adjust adiabatically to the motion along s, thus giving rise to the following effective potential:
VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA |
253 |
|
In the above equation |
is the array of conserved quantum numbers for |
|
the SA modes, and |
(neglected in this study) accounts for anharmonic effects and non |
|
orthogonality between the path tangent and the energy gradient [16,23]. In fact, the so called intrinsic reaction path (IRP) is always parallel to the gradient, so that the last contribution vanishes [23]. For intramolecular dynamics, however, the distinguished coordinate (DC) approach has the advantage of being isotope independent and well defined also beyond energy minima, while still retaining an almost negligible coupling between the gradient and the path tangent. This model corresponds to the construcion of the onedimensional path through the optimization of all the other geometrical parameters at selected values of a specific internal coordinate. In the present context, the distinguished coordinate is the out-of-plane angle defined in Figure 1. Furthermore, the distance s along the path is set to zero at a suitable reference configuration (in the present case the planar structure where
When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respective reference axes. The problem of translation has the trivial solution of centering the reference axes at the center of mass of the system. On the other hand, for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numerical minimization of the distance between successive points as a function of the Euler angles of the system [16,24].
In the scenario just sketched, the large amplitude vibration along s is governed by the following equation:
where |
is the kinetic energy operator and |
is a generic vibrational eigenstate with |
energy |
The so-called vibrationally adiabatic zero curvature (VAZC) approximation is |
|
obtained neglecting the small couplings between the path tangent and the local vibrational coordinates appearing in the kinetic energy operator [23,25]. Since the arc length is
measured in the space of mass weighted cartesian coordinates, we obtain a Schrödinger
254 V. BARONE ET AL.
equation formally equivalent to that governing the motion of a particle with a unit mass in a one dimensional space. This model is intimately linked to the use of local vibrational basis functions centered at different points along the path. In our approach [16,26,27], cubic splines are used to interpolate the potential along the path and to generate a larger set of equispaced points on which cubic splines are also used as basis functions. This kind of treatment avoids any modelling of the ab-initio data and involves only analytical integrals.
Although the size of the basis set is larger than the one necessary when employing Hermite, Morse or Gaussian functions, the spline approach remains competitive since the matrices to be diagonalized are banded with a constant width of 7. Furthermore, no new integrals are introduced by the computation of expectation values of observables (also represented by spline fittings), and the additional computational effort depends on the number of eigenstates to be taken into account, rather than on the dimension of the primitive spline
basis set. The expectation value |
of a given observable in the eigenstate j |
corresponding to the eigenvalue is given by |
|
The temperature dependence of the observable is obtained by assuming a Boltzmann population of the vibrational levels, so that
3. Results
Full geometry optimizations and calculations of harmonic force constants were performed at the UMP2/6-311G** level. Although this is not the main concern of this study, it is noteworthy that the relatively unexpansive theoretical treatment we have developped provides structural and spectroscopic parameters in close agreement with experiment (see Table 1). More precisely, the harmonic approximation seems quite adequate for whereas strong anharmonicities affect the CH stretchings and the out-of-plane motion of
The wave number of this latter vibration is increased to |
(in much better |
|
agreement with the experimental value of |
by our one-dimensional anharmonic |
|
treatment. Such a strong positive correction is in agreement with experimental estimates [29]. From another point of view, the two radicals are well suited to point out the influence
of the shape of the potential well on vibrational effects: a simple well |
and a double- |
well with an high inversion barrier |
|
The influence of out-of-plane bending on geometrical parameters, electronic energy and coupling constants is shown in Figures 2-4. The linear relationship between the s coordinate and the angle is well evidenced in Figure 2a. We recall that in our approach, although the geometries in internal coordinates used to build the path are mass independent, the arc length s varies with the atomic masses, whereas the reduced mass governing the motion always remains unitary. The larger mass of fluorine versus hydrogen then explains
the lower slope of the curve versus s for |
than for |
Also noteworthy is the |
increase of the CH and CF bond lengths upon inversion (Figure 2b).
VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA |
255 |
The symmetry assignment of vibrational states refers to |
point group. Experimental |
||
geometries and wave numbers are taken from [28,29] for |
and [30] for |
EPR |
|
parameters are taken from [31] for |
at 96K and [32] for |
.at77K. |
|
It is quite apparent (Figures 3,4) that the hyperfine constants of the central and terminal atoms in the two radicals are strongly influenced by the out-of-plane displacement. For the
central atom, the coupling increases with |
and this is clearly related to a strong change in |
hybridization. |
|