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Relationship between average and equilibrium values of internuclear distances in linear triatomic molecules of AB2 type
Journal of Molecular Structure, 76 (1981) 105-108 THEOCHEM Elsevier Scientific Publishing Company, Amsterdam
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Short communication
RELATIONSHIP BETWEEN AVERAGE AND EQUILIBRIUM OF INTERNUCLEAR DISTANCES IN LINEAR TRIATOMIC MOLECULES OF AB2 TYPE
V. I. BAZHANOV Institute
of High
(Received
VALUES
and V. V. KASPAROV Temperatures,
6 November
Korovinskoye
Rd.
127412,
Moscow
I-412
(C1.S.S.R.)
1979)
One of the most important characteristics of a molecule’s potential energy surface is its set of equilibrium internuclear distances, Re. These internuclear distances cannot be determined by gas electron diffraction but the average distance values, Rg, can be measured. The problem then is the conversion of Rg to R”. Recently, for the estimation of R”, investigators have been using the Ra value [l] calculated from Rg with allowance for the supposed probability density of internuclear distance distribution
R” = Rg - ( +)/2Rg
(1)
where < . . . . > denotes averaging over the probability density, and and are the mean-square perpendicular amplitudes of vibrations. For the case of linear triatomic molecules of the type AB, , eqn. (1) may be re-written as Rff(A-B)
= Rg(A-B)
where 6 (B-B)
- 6(B-B)/2
is the experimental
(2) value of the B-B distance shrinkage. The data is related to R” by
R” value calculated from electron diffraction Rff
=
R” +
(3)
where AZ is the parallel variation of the distance. Assuming that negligible, i.e.
is
AzQA,
(4)
(where An is the absolute error of the internuclear
distance determination)
Re should be closer to R” than to Rg [ 21. However, the mean values of internuclear distance Rg(Mg-F) obtained [ 31 for the MgF2 molecule coincides (within experimental accuracy) with the equilibrium values of R’ (Mg--F) calculated [ 4, 5,6] by means of the MO/LCAO/SCF method with a variable internuclear distance. However, the R” value (which from
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106
eqn. (3) and formula (4) should be close to Re (Mg-F)) differs considerably from the equilibrium value. Moreover, analysis of the AB, molecular vibrations, with the nuclear displacements perpendicular to the molecular axis chosen as vibrational coordinates describing the nuclear displacements, shows that the shrinkage value is associated with the doubled increment of the A-B internuclear distance [ 71. Another model attributes the shrinkage to the B-B distance decrease [ 81. We shall now demonstrate that both these effects result from the high anharmonicity of the potential energy surface in Cartesian coordinates, and find relations between Re(A-B) and Rg(A-B). Our calculations assume that the potential energy of molecular vibrations in natural vibrational coordinates q,, qz, q3, q4 (used in the generally accepted manner [9]) may be represented as a quadratic form (harmonic approximation). The relations linking the mean internuclear distances with the mean nuclear displacements from their equilibrium positions, accurate to secondorder terms, are Rg(B-B)
= 2R”(B-B)
Rg(A-B)
= R”(A-B)
+
(5)
+
+ (
+)/2Re(A-B) (6)
Here, Az~_~ and AZ,_, are the parallel variations of the B-B and A-B distances and and are the perpendicular mean-square vibrational amplitudes of the A-B nuclear pairs. Equation (6) together with eqn. (1) can determine the closeness of Ra (A-B) to R” (A-B) when AZ is small. Thus, we shall first calculate AZ. Since for a linear triatomic molecule
+
= Re(A-B)
2Re(A-B)
(.+I:>
8
+ <4i>)
and Rg(B-B)
= 2Re(A-B)
+ 2
(8)
then Rg(A-B)
= Re(A-B)
+
+ Re(A-B)
(
+)/8
(9)
To express the Az*_a value in terms of natural coordinates it is necessary to equate the instantaneous values of the R(A-B) distance expressed in natural coordinates to those in Cartesian coordinates (Re(A-B)
+ q1)2 = Ax~-~
Equations
(7) and (10) give Az*_a
AZ,_,
= [Re(A-B)
+ ql]
+ Ayi-a
l-
+ [AZ,_,
Ax;-a 2Re(A-B)
+ Re(A-B)]’
(16)
+ AY;-~
[l + qI/Re(A-B)12
- Re(A-B) I
(11)
107
Taking into account the dependence of Ax*_~ and AyA-a on natural coordinates and the mutual independence of Ax*_* and AyA-a and after averaging of eqn. (11) we obtain for the approximation of harmonic vibrations
= - Re(;-B)
(
+)
(12)
Noting that R”(A-B)
(
+)/4
and taking into account Rg(B-B)
= BR”(B-B)
Rg(A-B)
= Re(A-B)
= 6(B-B)
eqns. (8)) (9) and (12) - 6(B-B)
we obtain (13) (14)
Equations (13) and (14) are analogous to the respective expressions obtained for the natural vibrational coordinates (see, e.g. ref. 10). Thus, eqns. (5) and (6) derived for Cartesian coordinates give the same result as obtained with the analysis of the vibrations in natural coordinates, namely, that the shrinkage is associated with the B-B distance decrease. The conclusion on effective increase of the A-B distance is related to an assumption of low anharmonicity of the potential energy function in the Cartesian system (eqn. 4), which contradicts eqn. (12). Thus, if the potential energy function in the natural coordinate system is expressed in a quadratic form, then the potential energy in the Cartesian one is anharmonic and hence the Re(A-B) value is close to Rg(A-B) but not R&(A-B). Harmonicity of the potential energy function in the system of natural coordinates may be confirmed by comparison of the Rg values obtained from the electron-diffraction data with those of Re calculated in quantum chemistry. Maximum differences between Rg and R” should be observed for molecules characterized by large displacements of the nuclei from their equilibrium positions. In the latter case the difference between Rg and Re should exceed the errors in Rg measurement and those of quantum chemistry calculations. Therefore, we shall analyse the results of a high-temperature electron-diffraction study of alkaline-earth dihalides [ 31 which are characterized by low frequency bending vibrations (column 1 in Table 1). For MgF,, the Rg and R” values may be compared with the results of quantum chemistry calculations [ 4-61 presented in columns 4-6 in Table 1. The agreement between Rg and R” confirms the assumption of potential energy function harmonicity in normal coordinates since in this case Rg(A-B) is close to R”(A-B) (eqn. 14). Formulae (13) and (14) permit the calculation (within the framework of usual approximations [ 111) the equilibrium internuclear distances for linear AB2 molecules starting from the electron diffraction data. These formulae suggest the conclusion that a better estimate for Re( A-B) is Rg(A-B)
108 TABLE
1
Internuclear Molecule
distances
Internuclear
Cal, Sri,
in the molecules
distance
[31
R(A-B)
MgF, , MgCl,,
CaI,
and Sri,
(a)
R” [31
Re [41
Re 151
Re [61
1.771 + 0.010 2.185 k 0.011 2.867 * 0.015
1.718 2.129 2.758
1.77
1.762
1.77
3.009
2.864
Rg
MgF, WY&
R(A-B)
? 0.015
rather than Ra (A-B) calculated from eqn. (1). For the molecules MgF,, MgClz, CaI, and Sri, the values presented in column 1 of Table 1 should serve as such estimates. REFERENCES 1 K. Kuchitsu and S. J. Cyvin, in S. J. Cyvin (Ed.), Molecular Structure and Vibrations, Elsevier, Amsterdam, 1972, Ch. 12. 2. V. P. Spiridonov, in Sovremennie problemy fizitcheskoi khimii, v.9, Moscow University, Moscow, 1976. 3 Yu. Ezhov and V. Kasparov, 7th Austin Symposium on Gas Phase Molecular Structure, Austin, TX, U.S.A., 1978. 4 M. Astier, G. Berthier and P. Millie, J. Chem. Phys., 57 (1972) 5008. 5 J. L. Gole, A. K. Q. Siu and E. F. Hayes, J. Chem. Phys., 58 (1973) 857. 6 P. Pendergast and E. F. Hayes, J. Chem. Phys., 68 (1978) 4022. 7 Y. Morino and T. Iijima, Bull. Chem. Sot. Jpn., 35 (1962) 1661. 8 Y. Morino, J. Nakamura and P. W. Moore, J. Chem. Phys., 36 (1962) 1050. 9 V. M. Tatevsky, Stroyeniye molekul, Khimiya (Publishers), Moscow, 1977. 10 V. I. Bazhanov and V. V. Kasparov, J. Mol. Struct., in press. 11 Teoretitcheskiye osnovy gazovoy elektronografii, Moscow University, Moscow, 1974.
-
Printed
in The Netherlands
Short communication
RELATIONSHIP BETWEEN AVERAGE AND EQUILIBRIUM OF INTERNUCLEAR DISTANCES IN LINEAR TRIATOMIC MOLECULES OF AB2 TYPE
V. I. BAZHANOV Institute
of High
(Received
VALUES
and V. V. KASPAROV Temperatures,
6 November
Korovinskoye
Rd.
127412,
Moscow
I-412
(C1.S.S.R.)
1979)
One of the most important characteristics of a molecule’s potential energy surface is its set of equilibrium internuclear distances, Re. These internuclear distances cannot be determined by gas electron diffraction but the average distance values, Rg, can be measured. The problem then is the conversion of Rg to R”. Recently, for the estimation of R”, investigators have been using the Ra value [l] calculated from Rg with allowance for the supposed probability density of internuclear distance distribution
R” = Rg - ( +
(1)
where < . . . . > denotes averaging over the probability density, and
= Rg(A-B)
where 6 (B-B)
- 6(B-B)/2
is the experimental
(2) value of the B-B distance shrinkage. The data is related to R” by
R” value calculated from electron diffraction Rff
=
R” +
(3)
where AZ is the parallel variation of the distance. Assuming that
is
AzQA,
(4)
(where An is the absolute error of the internuclear
distance determination)
Re should be closer to R” than to Rg [ 21. However, the mean values of internuclear distance Rg(Mg-F) obtained [ 31 for the MgF2 molecule coincides (within experimental accuracy) with the equilibrium values of R’ (Mg--F) calculated [ 4, 5,6] by means of the MO/LCAO/SCF method with a variable internuclear distance. However, the R” value (which from
~166-1~~~/~1/~~~~_0~~0/$~2.50
0 1981
Elsevier
Scientific
Publishing
Company
106
eqn. (3) and formula (4) should be close to Re (Mg-F)) differs considerably from the equilibrium value. Moreover, analysis of the AB, molecular vibrations, with the nuclear displacements perpendicular to the molecular axis chosen as vibrational coordinates describing the nuclear displacements, shows that the shrinkage value is associated with the doubled increment of the A-B internuclear distance [ 71. Another model attributes the shrinkage to the B-B distance decrease [ 81. We shall now demonstrate that both these effects result from the high anharmonicity of the potential energy surface in Cartesian coordinates, and find relations between Re(A-B) and Rg(A-B). Our calculations assume that the potential energy of molecular vibrations in natural vibrational coordinates q,, qz, q3, q4 (used in the generally accepted manner [9]) may be represented as a quadratic form (harmonic approximation). The relations linking the mean internuclear distances with the mean nuclear displacements from their equilibrium positions, accurate to secondorder terms, are Rg(B-B)
= 2R”(B-B)
Rg(A-B)
= R”(A-B)
+
(5)
+
+ (
+
Here, Az~_~ and AZ,_, are the parallel variations of the B-B and A-B distances and
+
= Re(A-B)
2Re(A-B)
(.+I:>
8
+ <4i>)
and Rg(B-B)
= 2Re(A-B)
+ 2
(8)
then Rg(A-B)
= Re(A-B)
+
+ Re(A-B)
(
+
(9)
To express the Az*_a value in terms of natural coordinates it is necessary to equate the instantaneous values of the R(A-B) distance expressed in natural coordinates to those in Cartesian coordinates (Re(A-B)
+ q1)2 = Ax~-~
Equations
(7) and (10) give Az*_a
AZ,_,
= [Re(A-B)
+ ql]
+ Ayi-a
l-
+ [AZ,_,
Ax;-a 2Re(A-B)
+ Re(A-B)]’
(16)
+ AY;-~
[l + qI/Re(A-B)12
- Re(A-B) I
(11)
107
Taking into account the dependence of Ax*_~ and AyA-a on natural coordinates and the mutual independence of Ax*_* and AyA-a and after averaging of eqn. (11) we obtain for the approximation of harmonic vibrations
= - Re(;-B)
(
+
(12)
Noting that R”(A-B)
(
+
and taking into account Rg(B-B)
= BR”(B-B)
Rg(A-B)
= Re(A-B)
= 6(B-B)
eqns. (8)) (9) and (12) - 6(B-B)
we obtain (13) (14)
Equations (13) and (14) are analogous to the respective expressions obtained for the natural vibrational coordinates (see, e.g. ref. 10). Thus, eqns. (5) and (6) derived for Cartesian coordinates give the same result as obtained with the analysis of the vibrations in natural coordinates, namely, that the shrinkage is associated with the B-B distance decrease. The conclusion on effective increase of the A-B distance is related to an assumption of low anharmonicity of the potential energy function in the Cartesian system (eqn. 4), which contradicts eqn. (12). Thus, if the potential energy function in the natural coordinate system is expressed in a quadratic form, then the potential energy in the Cartesian one is anharmonic and hence the Re(A-B) value is close to Rg(A-B) but not R&(A-B). Harmonicity of the potential energy function in the system of natural coordinates may be confirmed by comparison of the Rg values obtained from the electron-diffraction data with those of Re calculated in quantum chemistry. Maximum differences between Rg and R” should be observed for molecules characterized by large displacements of the nuclei from their equilibrium positions. In the latter case the difference between Rg and Re should exceed the errors in Rg measurement and those of quantum chemistry calculations. Therefore, we shall analyse the results of a high-temperature electron-diffraction study of alkaline-earth dihalides [ 31 which are characterized by low frequency bending vibrations (column 1 in Table 1). For MgF,, the Rg and R” values may be compared with the results of quantum chemistry calculations [ 4-61 presented in columns 4-6 in Table 1. The agreement between Rg and R” confirms the assumption of potential energy function harmonicity in normal coordinates since in this case Rg(A-B) is close to R”(A-B) (eqn. 14). Formulae (13) and (14) permit the calculation (within the framework of usual approximations [ 111) the equilibrium internuclear distances for linear AB2 molecules starting from the electron diffraction data. These formulae suggest the conclusion that a better estimate for Re( A-B) is Rg(A-B)
108 TABLE
1
Internuclear Molecule
distances
Internuclear
Cal, Sri,
in the molecules
distance
[31
R(A-B)
MgF, , MgCl,,
CaI,
and Sri,
(a)
R” [31
Re [41
Re 151
Re [61
1.771 + 0.010 2.185 k 0.011 2.867 * 0.015
1.718 2.129 2.758
1.77
1.762
1.77
3.009
2.864
Rg
MgF, WY&
R(A-B)
? 0.015
rather than Ra (A-B) calculated from eqn. (1). For the molecules MgF,, MgClz, CaI, and Sri, the values presented in column 1 of Table 1 should serve as such estimates. REFERENCES 1 K. Kuchitsu and S. J. Cyvin, in S. J. Cyvin (Ed.), Molecular Structure and Vibrations, Elsevier, Amsterdam, 1972, Ch. 12. 2. V. P. Spiridonov, in Sovremennie problemy fizitcheskoi khimii, v.9, Moscow University, Moscow, 1976. 3 Yu. Ezhov and V. Kasparov, 7th Austin Symposium on Gas Phase Molecular Structure, Austin, TX, U.S.A., 1978. 4 M. Astier, G. Berthier and P. Millie, J. Chem. Phys., 57 (1972) 5008. 5 J. L. Gole, A. K. Q. Siu and E. F. Hayes, J. Chem. Phys., 58 (1973) 857. 6 P. Pendergast and E. F. Hayes, J. Chem. Phys., 68 (1978) 4022. 7 Y. Morino and T. Iijima, Bull. Chem. Sot. Jpn., 35 (1962) 1661. 8 Y. Morino, J. Nakamura and P. W. Moore, J. Chem. Phys., 36 (1962) 1050. 9 V. M. Tatevsky, Stroyeniye molekul, Khimiya (Publishers), Moscow, 1977. 10 V. I. Bazhanov and V. V. Kasparov, J. Mol. Struct., in press. 11 Teoretitcheskiye osnovy gazovoy elektronografii, Moscow University, Moscow, 1974.