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High-temperature inorganic molecular species with polytopic bonds
Structure, 28 (1975) 89-96 ~Eisevier Scientific Publishing Company, Amsterdam -
89
Journal of Molecular
Printed in The Netherlands
HIGH-TEMPERATURE INORGANIC MOLECULAR POLYTOPlC BONDS II. A PROPOSED DEFINITION OF POLYTOPICITY
SPECIES WITH
N. G. RAMBIDI* Institute of E&h Tempemture, U.S.S.R. Academy of Science, Korovinskoye Shosse, Moscow 12?412 (U.S.S.R.) and Department of Chemistry, Indiana University, Bloomington, Indiana (U.S.A.)
(Received 22 October 1974)
ABSTRACT A description of the nuclear distributions in high-temperature inorganic molecular species corresponding to different excited vibrational states was considered. The concept of the most probable structure was revised and used for the case of inorganic molecules with polytopic bonding character. INTRODUCTION
Molecules of complex inorganic compounds, with bonds of a polytopic nature [l] containing different relative positions of fragments which are unusually close considering the total electronic energy of the system present complex dynamic systems with large displacements of rather rigid fragments relative to each other. For the ca.seof the simplest of molecules considered, the description of structure was analyzed from the viewpoint of peculiarities of the surface of potential energy of the nuclei. It was shown that even using a very approximate model 'one was able to understand some essential and unusual details of the molecular structure of compounds of low-volatility. Nevertheless, such a description becomes complicated and loses the clarity of representation if the number of interacting molecular fragments is more than two_ LOW- AND HIGfi-TEMPERATURE
MOLECULAR
SPECIES
As is known, the basic concept of modem structural chemistry is that a unique equilibrium geometric configuration of nuclei, corresponding to the minimum of potential energy of the molecule, adequately describes the actual distribution of its nuclei. When the values of intramolecular energy differences are sufficiently great and the vapor temperature low (more accurately, if AE/kZ%l), this concept *Contribution No. 2607 from the Chemical Laboratories of Indiana Univer&v.
90
adequately describes the molecular structure. In other words, if the vapor contains molecules in which the electronic ground state and zero vibration levels predominantly are populated, the average and most probable values of internuclear distances differ only slightly from the equilibrium ones. (Classical rotational averaging is employed and it is assumed that centrifugal distortion effects are small compared with the vibrational features discussed here. It is important to remember that even if rotational effects are significant they will not affect the qualitative features of the vibrational probability distribution for a fixed value of the vibrational quantum numbers.) Thus, at low temperatures: (a) In vapor there usually exists a single structural species of the molecule considered, i.e. a molecule with a distribution of nuclei corresponding to the ground electronic and zero vibrational states, and (b) The actual distribution of nuclei of this structural species is close to a rigid equilibrium configuration of the molecule. At high vapor temperatures, in particular, if some vibration frequencies of the molecule are low (or the electronic state excitation energy is low), there may appear in the vapor structural species the actual distribution of whose nuclei may differ in principle from the equilibrium one. Let us consider, for example, the LiNC molecule which possesses a bending vibration frequency equal to 119 cm-’ [S] . In this case, at a sufficiently high temperature the vapor contains many molecules each corresponding to different degrees of vibrational excitation. Recall that the low-lying bending vibrations are usually the first to be thermally excited. Thus, at 2’ = 1500”K, the population of the 0,O level of a harmonic bending vibration with a frequency of 119 cm-’ is 1.16%, of the 0,l level 1.04% of the 1,l level 0.93%, while the summary population of the first levels (up to u = 20) is about 70%. In each vibrational level, the actual distribution density of the lithium nucleus is to a considerable degree spread out in space (Fig. 1). While so doing, only
.\\ 1
\
\
N \
‘.
C
:
------ A’
0 Fig. 1. Distribution functions of lithium atom in the LiNC molecule according to ref. 4. The horizontal axis shows variations of the angle q (see inset on right for defiiition) and the vertical axis indicates the potential energy on relative scale and also the vibrational probability distribution on an independent relative scale.
91
the zero level possesses a single maximum of probability density situated on the straight line interconnecting the nuclei of the nitrogen and carbon atoms. Therefore, it is only for the zero level of the bending vibration that the most probable configuration coincides with the symmetry type of the equilibrium species. For the overwhelming majority of structural species existing in marked concentration in vapor, the actual distribution of nuclei may differ considerably from the equilibrium one. Moreover, Clementi et al. [4] have shown that, upon the movement of the lithium atom around the NC grouping, the maximum value of the trough in the potential energy surface corresponds to the LiCN configuration and is only 0.014 atomic units higher than the value of energy at the point of equilibrium. Therefore, Clements et al. [4] assumed that in the LiNC molecule the lithium atom could undergo orbital motion around the NC group at very high temperatures. Generally speaking, this assumption does not seem obvious in the case of the LiNC molecule. It is not difficult to observe that even without taking into account a possible crowding of highly excited levels the total number of “bound” states of the bending vibration should be rather large. Therefore, any marked number of molecules exhibiting orbiting motion should only appear at temperatures about 1500-2000”K, i.e. in the region where thermal dissociation of the LiNC molecule is possible. Nevertheless, the probability of orbiting of the metal atom is certainly possible in the case of other molecules whose structural peculiarities are similar to those of LiNC. For these molecules, the very concept of rigid configuration, independent of the definition used, loses its meaning. It follows from the foregoing that at high temperatures: (a) The vapor contains appreciable amounts of several structural species of the molecule under consideration. Each species has a distribution of nuclei corresponding to the various energetic states of the molecule, and (b) The distribution of nuclei for the excited states of the molecule may differ from the equilibrium one even by symmetry type. In order to emphasize the difference between the cases of low-temperature and high-temperature vapor, let us consider one more example, namely molecular systems in which low-frequency vibrations of nuclei are described by a one-dimensional potential of the type
U(x) = 1 kc* +
A (c2
+
x2)
Here, the coordinate x: may characterize the deviation of the nuclear arrangement of a triatomic symmetrical molecule from the linear one 153, nonplanarity of a cyclic molecule of the C,H,El type where El = 0, S [6] and so on. It is not hard to see that in the case where the perturbation of the harmonic potential in eqn. (1) actually exists and in the case where it is absent the probability density functions for the same sufficiently highly excited level, containing the same number of nodal points, are close to each
92
other (Fig. 2). Therefore, even for different types of equilibrium configurations (angular and linear, planar and nonplanar), the actual distribution of nuclei for highly excited levels with the same number of nodal points may practically coincide. Therefore, based on the foregoing considerations, one can arrive at the conclusion that two obvious assertions underlie the structural ideas for hightemperature vapor molecules (and these, naturally, must include low-temperature vapor as a particular case), namely (a) for one and the same electronic state of a molecule one should distinguish between structural species present in the vapor in appreciable amounts, the actual distribution of whose nuclei corresponds to different vibrational states. These distributions may differ substantially from each other, and (b) in the event that a molecule contains a low-lying electronic level(s) it will be also necessary to specify the different electronic states. CONCEPT OF THE MOST PROBABLE
STRUCTURE
Specific features of the actual distribution of nuclei of a polyatomic molecule to which correspond different degrees of excitation of some normal vibrations have been discussed in detail in a monography by Vol’kenstein, et al. [7]. Quite naturally, in this case an accurate description appears complicated and the clarity of representation is rather poor. Therefore, it is natural, while retaining the concept of a rigid geometrical configuration, to try and introduce a configuration that would provide a possible means of characterizing the actual nuclear distribution in each of the various vibra-
U(x)
-B+
3(x)
+
Fig. 2. Distribution functions for higher excited levels of perturbed and unperturbed harmonic oscillators as a function of the nuclear displacement, X, and the vertical axis is used in the same manner as explained in Fig. 1.
93
tional states. Apparently best suited for this purpose is the most probable molecular configuration which may be constructed based on the most probable values of internuclear distances corresponding to the distances between the maximum points of the biggest (in the vicinity of the given nucleus) peaks on the surface of the distribution function of the nuclei. In order to clarify this definition, let us consider the simplest example of a diatomic molecule. As pointed out in ref. ‘7, upon transferring the origin of coordinates to the
center of mass of a molecule and introducing the Cartesian displacement coordinates ~6’) and x(‘) for the first and second nuclei, their relation to the normal coordinate Q = r - re will be expressed as
xc’ ) = -(M&Z)-
‘Q,
x(‘) = (M2M)-
‘Q,
M =
Mfy&
(2)
2
Therefore, the distribution function of the coordinates of each nucleus of the molecule reproduces the distribution function of the normal coordinate (Fig. 3). It can be readily seen that for each nucleus and for each vibrational state there exists one biggest peak of the distribution function. The distances between their maximum points for the various nuclei can then be taken as the most probable values of the internuclear distances. These distances increase monotonically with the vibrational quantum number, which is in agreement with intuitive assumptions on the dissociation of molecules in highly excited states. Starting with the first excited vibrational state, the distribution function for each nucleus contains, in addition to the maximum peak, a number of smaller peaks. Thus, the internuclear distance values actually possible for the given vibrational state may be distributed over a wide region in space.
Fig. 3. The distribution functions of the normal coordinate for a diatomic molecule (left side of figure) and the distribution functions of the two separate nuclei as a function of their respective displacement coordinates (right side) all vertical and horizontal scales are relative.
94
Therefore, it is natural to introduce an additional characteristic for each vibrational state, namely, degree of the structure polytopicity (degree to which the structure is polytopic). It can be found in a onedimensional case through the ratio of the area of the biggest peak of the distribution function to its total area DP = 1 - % P(x) dx/j
P(x) dx
(3)
where cuand p define the integration limits of the biggest peak. The degree of polytopicity of the structure, in the general case, must increase upon an increase of the vibrational quantum number. For the zero vibrational level its value is naturally zero. Upon free motion of nuclei in the region above the dissociative limit, the degree of polytopicity should be equal to unity. Threfore, in order to characterize the structure of a diatomic molecule in an excited vibrational state in the present approximation, one should know the most probable value of internuclear distance and degree of polytopicity of the structure. Generally speaking, each vibrational level should probably be regarded in comparison with its population at a given vapor temperature, which shows whether or not this structural species exists in the vapor. Quite naturally, in the case of big polyatomic molecules the description of the structure in terms of the most probable configuration introduced gets more complicated. Nevertheless, there is some reason to use it for inorganic complex molecules with bonds of polytopic nature (cf. ref. 1) in which a relatively small number of molecular fragments move with large amplitudes relative to each other. Let us take as an example a model of the CsN03 molecule, obtained as a result of electrostatic evaluation of the surface of the energy of interaction of cesium ion with the anion N03- for which the charge distribution of Z(0) = -0.55, Z(N) = 0.65 is assumed [I]. The results show that the equilibrium configuration of the nuclei for this model corresponds to a planar structure in which the cesium ion nucleus is on a straight line connecting the central atom nucleus and the middle of the O-O side of the N030 triangle to form the four-membered ring CsKo> N. At the same time, large displacements of the cesium ion in a plane normal to the plane of the NOBgroup are accompanied with very small variations of the total system energy (cf. Fig. 4). The curvature of the interaction energy along the direction of the valley bottom likewise turns out to be very small. Thus, in the CsN03 molecule there should exist a low-frequency out-ofplane vibration of the cesium ion and, consequently, a high population of excited vibration levels and large displacements of the cesium ion nucleus. Moreover, the frequencies of vibrations of the NO,- nuclei are much higher than those associated with cesium. Because of this, in the first approximation! it is possible to use for the description of the CsN03 molecule a model in which the cesium ion moves around the rigid N03- group as outlined below.
95
Fig. 4. First two contours of a cross section the potential energy surface for CsNO,, corresponding to 0.005 and 0.010 a.u. energy difference from the equilibrium energy point.
Apparently, in the first approximation the nature of the cesium ion movement in a direction along the valley bottom may be approximated by means of a one-dimensional harmonic oscillator. In this case, the maximum of the cesium nucleus distribution function for the zero vibration level will lie in the plane and, for the excited levels, above (and under) the plane of the NO,group. While so doing, the CsNO angle corresponding to the most probable configuration should decrease upon an increase in the degree of excitation of the bending vibration (Fig. 5). Thus, within the limits of the model under consideration the most probable configuration varies from a planar ring (zero vibrational state) towards a configuration of the trigonal pyramid type in which the cesium ion nucleus is above the central nitrogen atom of the N03group.
In the harmonic oscillator approximation, the degree of polytopicity of
the structure is 0.0 for the zero and first vibrational states corresponding to non-planar vibrations of the cesium nucleus in the CsN03 molecule and approximately equal to 0.2 for the second vibrational state and 0.3 for the third one. The CsN03 molecule is only one example of the large group of molecular
Fig. 5. Models for the most probable structurescorrespondingto first three excited levelsof the cesium atom vibrationalspectrum in the CsNO, molecule. The probability density
is on an arbitrary
scale.
96
species, i.e. inorganic complex compounds for which polytopicity may be one of the most important molecular features. These molecules are surprisingly stable components when they occur in the high-temperature vapors. Their behavior can be quite different from that exhibited by inorganic molecules which may exist in a low-temperature vapor phase [I]. The concept of polytopicity may well provide the correct explanation for the stability of these molecules at high temperatures and help us to understand their unusual properties. ACKNOWLEDGEMENTS
The author wishes to express his gratitude to Professor R. A. Bonham for reading the manuscript and helpful advise and to Mrs. Betty Grubb for her valuable help in the preparation of the manuscript. REFERENCES 1 N. G. Rambidi, J. Mol. Struct., to be published. 2 N. G. Rambidi and V. P. Spiridonov, Teplofiz. Vysokikh Temperature, Akad. Nauk SSSR, 2 (1964) 280. 3 Z. K. Ismail, R. H. Hauge and J. L. Margrase, J. C&em. Phys., 57 (1972) 5137.
4 5 6 7
E. Clementi, H. Kistenmacher and H. Popkie, J. Chem. Phys., 58 (1973) 2460. W. R. Thorson and I. Nakagawa, J. Chem. Phys., 33 (1960) 994. W. D. Gwinn and A. Luntz, Trans. Amer. Crystallogr. Assoc., 2 (1966) 90. M. V. Vol’kenstein, L. A. Gribov, M. A. El’yashevich and B. I. Stepanov, Kolebaniya Molecul (Molecular Vibrations), Moscow, 1973.
89
Journal of Molecular
Printed in The Netherlands
HIGH-TEMPERATURE INORGANIC MOLECULAR POLYTOPlC BONDS II. A PROPOSED DEFINITION OF POLYTOPICITY
SPECIES WITH
N. G. RAMBIDI* Institute of E&h Tempemture, U.S.S.R. Academy of Science, Korovinskoye Shosse, Moscow 12?412 (U.S.S.R.) and Department of Chemistry, Indiana University, Bloomington, Indiana (U.S.A.)
(Received 22 October 1974)
ABSTRACT A description of the nuclear distributions in high-temperature inorganic molecular species corresponding to different excited vibrational states was considered. The concept of the most probable structure was revised and used for the case of inorganic molecules with polytopic bonding character. INTRODUCTION
Molecules of complex inorganic compounds, with bonds of a polytopic nature [l] containing different relative positions of fragments which are unusually close considering the total electronic energy of the system present complex dynamic systems with large displacements of rather rigid fragments relative to each other. For the ca.seof the simplest of molecules considered, the description of structure was analyzed from the viewpoint of peculiarities of the surface of potential energy of the nuclei. It was shown that even using a very approximate model 'one was able to understand some essential and unusual details of the molecular structure of compounds of low-volatility. Nevertheless, such a description becomes complicated and loses the clarity of representation if the number of interacting molecular fragments is more than two_ LOW- AND HIGfi-TEMPERATURE
MOLECULAR
SPECIES
As is known, the basic concept of modem structural chemistry is that a unique equilibrium geometric configuration of nuclei, corresponding to the minimum of potential energy of the molecule, adequately describes the actual distribution of its nuclei. When the values of intramolecular energy differences are sufficiently great and the vapor temperature low (more accurately, if AE/kZ%l), this concept *Contribution No. 2607 from the Chemical Laboratories of Indiana Univer&v.
90
adequately describes the molecular structure. In other words, if the vapor contains molecules in which the electronic ground state and zero vibration levels predominantly are populated, the average and most probable values of internuclear distances differ only slightly from the equilibrium ones. (Classical rotational averaging is employed and it is assumed that centrifugal distortion effects are small compared with the vibrational features discussed here. It is important to remember that even if rotational effects are significant they will not affect the qualitative features of the vibrational probability distribution for a fixed value of the vibrational quantum numbers.) Thus, at low temperatures: (a) In vapor there usually exists a single structural species of the molecule considered, i.e. a molecule with a distribution of nuclei corresponding to the ground electronic and zero vibrational states, and (b) The actual distribution of nuclei of this structural species is close to a rigid equilibrium configuration of the molecule. At high vapor temperatures, in particular, if some vibration frequencies of the molecule are low (or the electronic state excitation energy is low), there may appear in the vapor structural species the actual distribution of whose nuclei may differ in principle from the equilibrium one. Let us consider, for example, the LiNC molecule which possesses a bending vibration frequency equal to 119 cm-’ [S] . In this case, at a sufficiently high temperature the vapor contains many molecules each corresponding to different degrees of vibrational excitation. Recall that the low-lying bending vibrations are usually the first to be thermally excited. Thus, at 2’ = 1500”K, the population of the 0,O level of a harmonic bending vibration with a frequency of 119 cm-’ is 1.16%, of the 0,l level 1.04% of the 1,l level 0.93%, while the summary population of the first levels (up to u = 20) is about 70%. In each vibrational level, the actual distribution density of the lithium nucleus is to a considerable degree spread out in space (Fig. 1). While so doing, only
.\\ 1
\
\
N \
‘.
C
:
------ A’
0 Fig. 1. Distribution functions of lithium atom in the LiNC molecule according to ref. 4. The horizontal axis shows variations of the angle q (see inset on right for defiiition) and the vertical axis indicates the potential energy on relative scale and also the vibrational probability distribution on an independent relative scale.
91
the zero level possesses a single maximum of probability density situated on the straight line interconnecting the nuclei of the nitrogen and carbon atoms. Therefore, it is only for the zero level of the bending vibration that the most probable configuration coincides with the symmetry type of the equilibrium species. For the overwhelming majority of structural species existing in marked concentration in vapor, the actual distribution of nuclei may differ considerably from the equilibrium one. Moreover, Clementi et al. [4] have shown that, upon the movement of the lithium atom around the NC grouping, the maximum value of the trough in the potential energy surface corresponds to the LiCN configuration and is only 0.014 atomic units higher than the value of energy at the point of equilibrium. Therefore, Clements et al. [4] assumed that in the LiNC molecule the lithium atom could undergo orbital motion around the NC group at very high temperatures. Generally speaking, this assumption does not seem obvious in the case of the LiNC molecule. It is not difficult to observe that even without taking into account a possible crowding of highly excited levels the total number of “bound” states of the bending vibration should be rather large. Therefore, any marked number of molecules exhibiting orbiting motion should only appear at temperatures about 1500-2000”K, i.e. in the region where thermal dissociation of the LiNC molecule is possible. Nevertheless, the probability of orbiting of the metal atom is certainly possible in the case of other molecules whose structural peculiarities are similar to those of LiNC. For these molecules, the very concept of rigid configuration, independent of the definition used, loses its meaning. It follows from the foregoing that at high temperatures: (a) The vapor contains appreciable amounts of several structural species of the molecule under consideration. Each species has a distribution of nuclei corresponding to the various energetic states of the molecule, and (b) The distribution of nuclei for the excited states of the molecule may differ from the equilibrium one even by symmetry type. In order to emphasize the difference between the cases of low-temperature and high-temperature vapor, let us consider one more example, namely molecular systems in which low-frequency vibrations of nuclei are described by a one-dimensional potential of the type
U(x) = 1 kc* +
A (c2
+
x2)
Here, the coordinate x: may characterize the deviation of the nuclear arrangement of a triatomic symmetrical molecule from the linear one 153, nonplanarity of a cyclic molecule of the C,H,El type where El = 0, S [6] and so on. It is not hard to see that in the case where the perturbation of the harmonic potential in eqn. (1) actually exists and in the case where it is absent the probability density functions for the same sufficiently highly excited level, containing the same number of nodal points, are close to each
92
other (Fig. 2). Therefore, even for different types of equilibrium configurations (angular and linear, planar and nonplanar), the actual distribution of nuclei for highly excited levels with the same number of nodal points may practically coincide. Therefore, based on the foregoing considerations, one can arrive at the conclusion that two obvious assertions underlie the structural ideas for hightemperature vapor molecules (and these, naturally, must include low-temperature vapor as a particular case), namely (a) for one and the same electronic state of a molecule one should distinguish between structural species present in the vapor in appreciable amounts, the actual distribution of whose nuclei corresponds to different vibrational states. These distributions may differ substantially from each other, and (b) in the event that a molecule contains a low-lying electronic level(s) it will be also necessary to specify the different electronic states. CONCEPT OF THE MOST PROBABLE
STRUCTURE
Specific features of the actual distribution of nuclei of a polyatomic molecule to which correspond different degrees of excitation of some normal vibrations have been discussed in detail in a monography by Vol’kenstein, et al. [7]. Quite naturally, in this case an accurate description appears complicated and the clarity of representation is rather poor. Therefore, it is natural, while retaining the concept of a rigid geometrical configuration, to try and introduce a configuration that would provide a possible means of characterizing the actual nuclear distribution in each of the various vibra-
U(x)
-B+
3(x)
+
Fig. 2. Distribution functions for higher excited levels of perturbed and unperturbed harmonic oscillators as a function of the nuclear displacement, X, and the vertical axis is used in the same manner as explained in Fig. 1.
93
tional states. Apparently best suited for this purpose is the most probable molecular configuration which may be constructed based on the most probable values of internuclear distances corresponding to the distances between the maximum points of the biggest (in the vicinity of the given nucleus) peaks on the surface of the distribution function of the nuclei. In order to clarify this definition, let us consider the simplest example of a diatomic molecule. As pointed out in ref. ‘7, upon transferring the origin of coordinates to the
center of mass of a molecule and introducing the Cartesian displacement coordinates ~6’) and x(‘) for the first and second nuclei, their relation to the normal coordinate Q = r - re will be expressed as
xc’ ) = -(M&Z)-
‘Q,
x(‘) = (M2M)-
‘Q,
M =
Mfy&
(2)
2
Therefore, the distribution function of the coordinates of each nucleus of the molecule reproduces the distribution function of the normal coordinate (Fig. 3). It can be readily seen that for each nucleus and for each vibrational state there exists one biggest peak of the distribution function. The distances between their maximum points for the various nuclei can then be taken as the most probable values of the internuclear distances. These distances increase monotonically with the vibrational quantum number, which is in agreement with intuitive assumptions on the dissociation of molecules in highly excited states. Starting with the first excited vibrational state, the distribution function for each nucleus contains, in addition to the maximum peak, a number of smaller peaks. Thus, the internuclear distance values actually possible for the given vibrational state may be distributed over a wide region in space.
Fig. 3. The distribution functions of the normal coordinate for a diatomic molecule (left side of figure) and the distribution functions of the two separate nuclei as a function of their respective displacement coordinates (right side) all vertical and horizontal scales are relative.
94
Therefore, it is natural to introduce an additional characteristic for each vibrational state, namely, degree of the structure polytopicity (degree to which the structure is polytopic). It can be found in a onedimensional case through the ratio of the area of the biggest peak of the distribution function to its total area DP = 1 - % P(x) dx/j
P(x) dx
(3)
where cuand p define the integration limits of the biggest peak. The degree of polytopicity of the structure, in the general case, must increase upon an increase of the vibrational quantum number. For the zero vibrational level its value is naturally zero. Upon free motion of nuclei in the region above the dissociative limit, the degree of polytopicity should be equal to unity. Threfore, in order to characterize the structure of a diatomic molecule in an excited vibrational state in the present approximation, one should know the most probable value of internuclear distance and degree of polytopicity of the structure. Generally speaking, each vibrational level should probably be regarded in comparison with its population at a given vapor temperature, which shows whether or not this structural species exists in the vapor. Quite naturally, in the case of big polyatomic molecules the description of the structure in terms of the most probable configuration introduced gets more complicated. Nevertheless, there is some reason to use it for inorganic complex molecules with bonds of polytopic nature (cf. ref. 1) in which a relatively small number of molecular fragments move with large amplitudes relative to each other. Let us take as an example a model of the CsN03 molecule, obtained as a result of electrostatic evaluation of the surface of the energy of interaction of cesium ion with the anion N03- for which the charge distribution of Z(0) = -0.55, Z(N) = 0.65 is assumed [I]. The results show that the equilibrium configuration of the nuclei for this model corresponds to a planar structure in which the cesium ion nucleus is on a straight line connecting the central atom nucleus and the middle of the O-O side of the N030 triangle to form the four-membered ring CsKo> N. At the same time, large displacements of the cesium ion in a plane normal to the plane of the NOBgroup are accompanied with very small variations of the total system energy (cf. Fig. 4). The curvature of the interaction energy along the direction of the valley bottom likewise turns out to be very small. Thus, in the CsN03 molecule there should exist a low-frequency out-ofplane vibration of the cesium ion and, consequently, a high population of excited vibration levels and large displacements of the cesium ion nucleus. Moreover, the frequencies of vibrations of the NO,- nuclei are much higher than those associated with cesium. Because of this, in the first approximation! it is possible to use for the description of the CsN03 molecule a model in which the cesium ion moves around the rigid N03- group as outlined below.
95
Fig. 4. First two contours of a cross section the potential energy surface for CsNO,, corresponding to 0.005 and 0.010 a.u. energy difference from the equilibrium energy point.
Apparently, in the first approximation the nature of the cesium ion movement in a direction along the valley bottom may be approximated by means of a one-dimensional harmonic oscillator. In this case, the maximum of the cesium nucleus distribution function for the zero vibration level will lie in the plane and, for the excited levels, above (and under) the plane of the NO,group. While so doing, the CsNO angle corresponding to the most probable configuration should decrease upon an increase in the degree of excitation of the bending vibration (Fig. 5). Thus, within the limits of the model under consideration the most probable configuration varies from a planar ring (zero vibrational state) towards a configuration of the trigonal pyramid type in which the cesium ion nucleus is above the central nitrogen atom of the N03group.
In the harmonic oscillator approximation, the degree of polytopicity of
the structure is 0.0 for the zero and first vibrational states corresponding to non-planar vibrations of the cesium nucleus in the CsN03 molecule and approximately equal to 0.2 for the second vibrational state and 0.3 for the third one. The CsN03 molecule is only one example of the large group of molecular
Fig. 5. Models for the most probable structurescorrespondingto first three excited levelsof the cesium atom vibrationalspectrum in the CsNO, molecule. The probability density
is on an arbitrary
scale.
96
species, i.e. inorganic complex compounds for which polytopicity may be one of the most important molecular features. These molecules are surprisingly stable components when they occur in the high-temperature vapors. Their behavior can be quite different from that exhibited by inorganic molecules which may exist in a low-temperature vapor phase [I]. The concept of polytopicity may well provide the correct explanation for the stability of these molecules at high temperatures and help us to understand their unusual properties. ACKNOWLEDGEMENTS
The author wishes to express his gratitude to Professor R. A. Bonham for reading the manuscript and helpful advise and to Mrs. Betty Grubb for her valuable help in the preparation of the manuscript. REFERENCES 1 N. G. Rambidi, J. Mol. Struct., to be published. 2 N. G. Rambidi and V. P. Spiridonov, Teplofiz. Vysokikh Temperature, Akad. Nauk SSSR, 2 (1964) 280. 3 Z. K. Ismail, R. H. Hauge and J. L. Margrase, J. C&em. Phys., 57 (1972) 5137.
4 5 6 7
E. Clementi, H. Kistenmacher and H. Popkie, J. Chem. Phys., 58 (1973) 2460. W. R. Thorson and I. Nakagawa, J. Chem. Phys., 33 (1960) 994. W. D. Gwinn and A. Luntz, Trans. Amer. Crystallogr. Assoc., 2 (1966) 90. M. V. Vol’kenstein, L. A. Gribov, M. A. El’yashevich and B. I. Stepanov, Kolebaniya Molecul (Molecular Vibrations), Moscow, 1973.