Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf256 |
V. BARONE ET AL. |
The hyperfine coupling constants of the hydrogens increase smoothly (in absolute value) with inversion, while those of the fluorines show a more complex trend, reaching their maximum value around the equilibrium structure. At his stage, and at the repective
equilibrium geometries, the couplings (Table 1) are far from experiment for |
but closer |
|
to experiment for |
More generally, the difference between computed and experimental |
|
values is inversely proportional to the height of the potential barrier, the effect being more pronounced at the central atom than at the α ones [33].
4. Discussion
The similarity in the behaviour of coupling constants as a function of in both radicals allows to discuss vibrational averaging effects simply in terms of the potential governing
the out-of-plane motion.
VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA |
257 |
The ground vibrational wave function of planar systems is peaked at the planar structure. Vibrational averaging then changes the coupling constants toward values which
would be obtained for an angle |
in a static description. The wave function of the |
|
ground vibrational state being symmetrically spread around |
introducescontributions |
|
of pyramidal configurations. This results in a noticeable increase of the absolute values of the coupling constants, which are minimal at planar structures (see Figure 3). Vibrational averaging then provides hyperfine coupling constants in close agreement with experiment. The effect is even more pronounced in the first excited vibrational state, whose wave function has a node at the planar structure and is more delocalized than the fundamental one, thus giving increased weight to pyramidal structures.
For radicals characterized by a double-well potential the vibrational effect acts in an opposite direction, bringing the coupling constants to values which would be obtained for The ground state vibrational wave function is now more localized inside the potential well, even under the barier, than outside. So it introduces more contributions of internal points. Vibrational effects, while still operative, are less apparent in this case since high energy barriers imply high vibrational frequencies with the consequent negligible population of excited vibrational states and smaller displacements around the equilibrium positions. This explains the good agreement between experimental and static theoretical computations.
Let us now turn to the second parameter, namely the shape of the "property surface". Around reference configurations, the dependence of the hyperfine coupling constants on the inversion motion is well represented by:
The average value of a can be written as:
The mean and mean square values of the LA coordinate s represent the principal anharmonic and harmonic vibrational contributions, respectively [3].
In the case of a planar equilibrium structure, the lineaar term is absent since symmetry
constraints impose that Since, in our case, hyperfine coupling
constants reach a minimum value at the planar reference structure (Figures 3b and 4b), the third term is always positive. Vibrational frequencies of this class of molecules are, of course small (Table 1), leading to large mean square amplitudes and consequently, to significant corrections to static values computed at the reference structure.
Unless Boltzmann averaging gives significant weight to vibrational states above the barrier,
strongly pyramidal molecules like |
can be effectively treated as systems governed by a |
single well potential unsymmetrically rising on the two sides of the minimum energy configuration. If we shift s so that now s = 0 at the equilibrium structure, the difference
with the previous case resides in the presence of the linear term in Eq.(8). This is due to the
258 |
V. BARONE ET AL. |
lack of any constraint on |
and < s >. Although the linear term contributes to < a >, it |
is, anyway, small and, since in our case < s > and the first derivative of coupling constants have opposite signs (see appendix), it conterbalances the harmonic contribution. Thus the resulting correction on < a > is small in all cases. this explains why the static results are very close to the dynamic ones.
5. Summary and conclusion
The results presented in the preceding sections call for the following general remarks.
i)As noticed in the earliest works on EPR [27,28], all the coupling constants increase, in absolute value, with the pyramidality at the radical center, the effect being always much prounounced at the radical enter than at the surrounding atoms.
ii)Vibrational averaging of coupling constants is always operative, but can be masked by the compensation of effects related to the shape of the potential energy surface from one side, and of the "property surface" from the other.
iii)From a methodological point of view, standard polarized basis sets and limited CI are
sufficient to compute hyperfine coupling constantsof localized -radicals, if large amplitude vibrations are properly taken into account.
The most significant outcome of our study is that a qualitative understanding of vibrational averaging effects is possible along the line of reasoning developed above. This opens the opportunity for a more dynamically based analysis of EPR parameters for large non rigid radicals.
Acknowledgments
The work of V.B. and C.M. was sponsored by the Italian Research Council (CNR Comitato Informatica), whose support is gratefully acknowledged.
Appendix
Using second order perturbation theory [3], the mean and mean square values of the mass weighted coordinate s in the vibrational state with quantum number j are explicitely given by:
where |
is the harmonic angular frequency |
VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA |
259 |
In the above equation, h is the Planck constant, and c the speed of light. The mean values at the absolute temperature T are obtained from the same equations by the of (j + 1/2) by
where K is the Boltzmann constant.From one side, Eq. (Al) shows that < s > and the cubic
force constants |
have opposite signs. On the other side, Figure 4 shows that in the |
|
case of |
and |
have the same sign near the equilibrium structure. As a |
result, the linear term in Eq. (8) is negative, thus counterbalancing the positive quadratic term.
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Ab–initio Calculations of Polarizabilities in Molecules: Some Proposals to this Challenging Problem
M. TADJEDDINE, J.P. FLAMENT
Ecole polytechnique, D.C.M.R., 91128 Palaiseau Cedex, France
1.Introduction
The polarizability expresses the capacity of a system to be deformed under the action of electric field : it is the first–order response. The hyperpolarizabilities govern the non linear processes which appear with the strong fields. These properties of materials perturb the propagation of the light crossing them; thus some new phenomenons (like second harmonic and sum frequency generation) appear, which present a growing interest in instrumentation with the lasers development. The necessity of prediction
of these observables requires our attention. |
|
The calculation of the static polarizability, |
is now well documented, actually com- |
mon enough to give a test for the choice of the atomic basis sets in molecular calculations. On the other hand, few calculations concern the dynamic polarizability, i.e. when the energy of the electric field is no more zero but can vary and reach the electronic transition energies of the molecule. Computations are more complex; not only they must well describe the ground state in order to reproduce the static polarizability, but also the excited states (valence and Rydberg states) in order to give the resonance energies correctly.
The computation of these observables poses several problems :
• In the case of an electromagnetic perturbation, a first difficulty rises : the choice
of the gauge. Indeed the gauge is only a mathematical tool and the observables of interest (energies, susceptibilities...) must be gauge invariant; they are effectively if the computations use complete molecular bases. Our calculations, using bases unavoidably truncated, will be never gauge invariant. The discrepancies with respect to the gauge invariance is, in a way, a mesure of the quality of our computation, of the molecular basis set. In our calculations
the gauge |
is used. |
•Since we must restrict the number,N, of the molecular states used in the com- putations, what value have we to give to N ? And then, can we correct the obtained value for the polarizability in order to approximate to the exact value by evaluating the ignored terms ?
•The formula which gives the polarizability involves the excited states. As said before, it is necessary to be able to well describe them. The choice of the atomic
261
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 261–278.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
262 |
M. TADJEDDINE AND J. P. FLAMENT |
basis set is essential : the bases used in usual calculations are not sufficient; we have to find suitable bases.
The first part of this paper responds to the first two problems through the calculation of the polarizability of CO (1). In this work, we bring our contribution to the three formal challenges enumerated by Ratner (2) in the special issue of Int. J. Quant.
Chem. devoted to the understanding and calculation of the non linear optical response of molecules :
1.The frequency dependence is taken into account through a ”mixed” time– dependent method which introduces a dipole–moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF–CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum–over– states.
2.The effects of electron correlation are investigted through the CIPSI (Configuration Interaction with Perturbatively Selected Configurations) calculations (4) of the molecular states.
3.The vibronic coupling features are evaluated in a perturbation treatment by taking account of temperature and electric field dependence (5).
The second part of this paper concerns the choice of the atomic basis set and especially
the polarization functions for the calculation of the polarizability, |
and the hyperpo- |
|
larizabiliy, |
We propose field–induced polarization functions (6) |
constructed from |
the first– and second–order perturbed hydrogenic wavefunctions respectively for and In these polarization functions the exponent is determined by optimization with the maximum polarizability criterion. These functions have been successfully
applied to the calculation of the polarizabilities, |
and |
for the He, Be and Ne |
|
atoms and the |
molecule. |
|
|
Throughout, atomic units will be used :
The unit of the dipole moment is equal to the unit of the dipole polarizability is equal to
and that of the second hyperpolarizability to
2.Calculation of the dynamic polarizability of CO : exemple of a mixed method
2.1.THEORY
2.1.1. The sum–over–states approach
The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrödinger equation, restricted to its electronic part and
the electric dipole interaction regime. The |
Stark Hamiltonian |
describes the |
dipolar interaction between the electric field |
and the molecule represented by its |
|
AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES |
263 |
dipole moment |
The perturbed molecular wavefunction is expanded in terms of the |
|
complete set of eigenfunctions |
of the unperturbed molecular Hamiltonian |
|
and the components of the polarizability are given through an expansion over all electronic excited states. For a static external electric field,
where u and v run over the cartesian electronic coordinates x, y and z. For an oscillating electromagnetic field characterized by its pulsation
the dynamic polarizability is derived from the time–dependent perturbation theory :
In such an expression, |
must be read as the sum of two functions |
like |
|
where |
|
The ket |
and its counterpart |
are calculated as a weighted sum over the |
|
excited states; the weight of each state is well defined through its interaction |
|||
with the |
ground state by the |
operator. The function |
represents |
the first–order perturbed wavefunction whose knowledge is essential in the variation– perturbation treatment. Expression (5) has been proposed by Karplus and Kolker
(7).
2.1.2. The polynomial approach
Previously, Kirkwood(8) had suggested another choice: he deduced the first–order perturbed wavefunction from the unperturbed one which was multiplied by a linear combination of the electronic coordinates, i.e. :
with : |
|
is the electric field component along v direction and |
some constants. In this |
approach, the polarizability may be calculated very easily |
from the second–order |
perturbed wavefunction which is simply given by :
264 |
M. TADJEDDINE AND J. P. FLAMENT |
For a heteronuclear diatomique molecule of |
symmetry (z being the molecular |
axis), it becomes : |
|
with :
where |
is the electron number. As tends to zero, the constants |
tend to |
The normalization condition |
imposes to move the origin to the center of |
electronic charge |
thus, the polarizability may be written very |
simply in the limit of zero frequency : |
|
2.1.3. A mixed approach
The idea to combine a method only polynomial (Eq.6 with |
and |
) with |
|
the SCF–CI procedure (Eq.5 with |
) has been initially developed for |
||
the calculation of magnetic observables (9) and later for the electric ones (10). Thus, the first–order perturbed wavefunction is given by :
and the component |
of the polarizability tensor becomes : |
The calculations of the and constants lead to a system of linear equations similar to that of the SCF–CI method, but with three more lines and columns corre- sponding to the coupling of the polynomial function with the electric field perturba- tion. The methodology and computational details have already been discussed (1); we stress two points : the role of the dipolar factor, the nature and the number of the excited states to include in the summation.
AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES |
265 |
2.2. DIPOLAR FACTOR
The dipolar factor |
may be interpreted in terms of gauge invariance. The electric |
||
observables usually are calculated in the gauge |
In the change to |
||
the gauge |
the Hamiltonian is transformed and the wavefunction |
becomes |
|
(11): |
|
|
|
If the strength of the electric field is small enough, then :
As known (11), the gauge invariance is ensured i f :
By omitting time–dependent terms, as in the preceding paragraph, the |
function |
|
may be read as the sum of the unperturbed wavefunction |
and a term which is the |
|
product of this function by a linear combination of the electronic coordinates, i.e. the
Kirkwood’s |
function. Thus, the |
dipolar factor ensures gauge–invariance. |
|
But the role of the dipolar factor |
in this mixed method is essential on the |
||
following point : its contribution in the |
computation occurs in a complementary |
||
(and sometimes preponderant) way to that calculated only from the |
excited states, |
||
the number of which is unavoidably limited by the computation limits. But before discussing their number, we have to comment the description of these states.
2.3. EXCITED STATES AND EXTRAPOLATION PROCEDURE
In a first approach, |
Rérat (10) described the |
excited states of Eq.15 |
through |
Slater determinants, |
constructed by monoexcitation of the ground |
state |
|
through the monoelectronic operator. By reason of orthogonality (deriving from all those necessary to the description of were rejected. The lack of such determinants does not allow to have a good description of the excited states
when they have a dominant configuration appearing also in If this approach led to interesting static results with reduced basis sets, it could not reach the resonances
correctly. |
|
|
It is the reason for which the Slater determinants have been replaced by the |
kets |
|
accounting for the true spectral states |
(1). These states have been computed |
|
independently by the CIPSI (4) program which treats the electronic correlation. Preliminary calculations of energies have been made by the standard CIPSI algorithm
(4a) on small S subspaces of c.a. 400 determinants. Perturbation treatments involving larger subspaces (about 1000 for CO) have been achieved using the diagrammatic version of CIPSI (4b).
The quality of the |
states has been tested through their |
energy and also their |
transition moment. |
Moreover from the natural orbitals and |
Mulliken populations |
analysis, we have determined the predominant electronic configuration of each
state and its Rydberg character. Such an analysis is particularly interesting since it explains the contribution of each to the calculation of the static or dynamic polarizability; it allows a better understanding in the case of the CO molecule : the difficulty of the calculation and the wide range of published values for the parallel
component while the computation of the perpendicular component is easier. In effect in the case of CO :