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Some properties of the potential energy surface and vibrational spectrum of a strong hydrogen bond complex
Journal of Molecular Structure, 177 (1988) 495-512
495
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
S O M E P R O P E R T I E S OF T H E P O T E N T I A L E N E R G Y S U R F A C E A N D V I B R A T I O N A L S P E C T R U M OF A S T R O N G H Y D R O G E N BOND COMPLEX
G.V. YUKHNEVICH and E.G. TARAKANOVA
N.S. Kurnakov Institute of General and Inorganic Chemistry, Academy of Sciences of the U.S.S.R., 31 Leninsky Avenue, Moscow 117071 (U.S.S.R.) (Received 14 September 1987)
ABSTRACT The forms of potential energy surfaces of weak and strong hydrogen bonds are analyzed. A method is proposed to take account of anharmonicity of vibrations of A - H - - . B hydrogen bridges. With this method are calculated the spectra of water dimer and of four complexes having symmetric hydrogen bonds. Good agreement between the computed and experimentally obtained spectra suggests that the proposed method can be adequately applied to compute vibrational spectra of systems containing hydrogen bonds.
INTRODUCTION
Since its postulation in the 1930s the hydrogen bond has been regarded as a purely specific form of interaction. Perhaps this is the reason why all its spectral manifestations have been explained by its individuality. With these manifestations were primarily placed the red shift of the stretching vibrations band of the O-H bond (VOH) and the weaker blue shift of the band of its bending vibrations (v~). Also, the gain in the intensities of VOHbands and their mutual location in the isotope-substituted molecules and in different OHx groups were thought to be typical exclusively for the hydrogen bond. The nature of most of the enumerated effects was understood in the 1960s when the semi-empirical theory of vibrations ofpolyatomic molecules was systematically applied to the complexes containing hydrogen bonds. All these effects were found to result from the phenomena well-known by that time in the spectroscopy of organic compounds. It was ascertained [ 1-3 ] that the spectral manifestations of a hydrogen bond were in principle the consequence of variations in the kinematics of the system and of the mutual effect of the neighbouring bonds. The more so, the assumption that a simple linear dependence exists between the dynamic parameters of the hydrogen bond made it possible to describe most of its then known spectral properties [2]. 0022-2860/88/$03.50
© 1988 Elsevier Science Publishers B.V.
496 Later a similar situation was shown to prevail in describing the intensities of absorption bands related to the vibrations of the hydrogen bridge [4 ]. Thus, in the 1960-1970s the whole body of theoretical and experimental data suggested that no additional or special assumptions were needed to describe the spectra of systems containing hydrogen bonds. Further development of spectral equipment made it possible to measure spectra in greater detail and to significantly widen the range of the systems studied. As a result, new spectral features of the systems containing hydrogen bonds were noticed, which could not be taken into account from the standpoint of the theory of harmonic vibrations. Thus, in the spectra of many hydrogenbonded liquid and amorphous systems continuous absorption caused by the vibrations of the X l " " - H . . "X2 fragment was noticed in the 3600-1000 cm -1 range [ 5, 6 ]. A detailed theoretical analysis of this phenomenon has revealed that continuous absorption cannot be caused by the medium's effect [7]. Slightly later, absorption in the spectra of water vapour was recorded in the 1200-800 cm-1 range where there can be none of the fundamental vibrations of water clusters [8, 9]. In the past few years, this absorption has frequently been related to the water dimers [10-12 ]. In both the examples the observed absorption is ascribed to the combination bands caused by vibrations including the hydrogen bond [ 12, 13 ]. As a check on this assumption it was necessary to quantitatively characterize the anharmonicity of hydrogen bonds of different strength. OBJECT OF INVESTIGATION In this paper an attempt has been made to provide a common system for describing the potential energy surface of complexes having hydrogen bonds. The coefficients of dynamic anharmonicity found by the proposed method were used to calculate the vibrational spectra of a number of systems having strong and weak hydrogen bonds. The effect of electro-optical and mechanical anharmonicity of vibrations on the nature of computed spectra has been analyzed. The calculated spectra have been compared with the experimental ones. This work was made possible by considerable progress made in quantum chemistry, the theory of anharmonic vibrations and computer facilities. The bases of the method of computing anharmonic vibrations in polyatomic molecules were laid in refs. 14 and 15. Therein it was proposed to perform direct numerical diagonalization of the vibrational part of the Hamiltonian using Ritz's variation method. The vibrational spectra considered below were computed by a set of programmes [ 16 ] where the basis set contained up to 10 000 harmonic oscillatory functions. Equilibrium configurations, force constants and the distribution of charges on the atoms of the studied molecules were found by using a set of programmes [ 17 ].
497 When this work was undertaken, information on the form of the potential surface of hydrogen bond was very scanty. The quantum-chemical studies were persued in a somewhat different direction. Much attention was paid to checking the correctness of equilibrium configurations computed for complexes having strong and weak hydrogen bonds [18]. In studying different interaction mechanisms it was shown that the donor-acceptor interaction played an insignificant role in the case of intermolecular and ion-molecular hydrogen bonds. The main contribution was made by the Coulomb and exchange forces [18]. Also, the potential curves of strong and weak hydrogen bonds were considered to have a completely different form. It was assumed that strong asymmetric hydrogen bonds had two minima and the decrease in the VOH vibration frequency was due to the dynamic interaction of A-H and H-- .B bonds of the bridge [ 19 ]. All this was in accord with the intuitive concept of the specific nature of a hydrogen bond. SOME DYNAMICCHARACTERISTICSOF A HYDROGENBOND Before we embark on the study of dynamic anharmonicity of hydrogen bonds, it is necessary to choose an analytical form for describing the potential energy surface and the parameters that would definitely reflect the extent to which the function deviates from the purely harmonic function and would be independent of the bond length and energy. The analytical expressions for the form of potential energy surface of a molecule may be diverse. In the literature, the form is more often described by a Taylor polynomial series, usually containing quadratic, cubic and quartic terms with coefficients F (R 2) , F (R 3) and F (R 4) , respectively. To facilitate working with the literature data, we shall also make use of this form of the expression. Obviously, the form of the potential energy surface of a hydrogen bridge, like any multidimensional surface, cannot be described by one parameter. Therefore, it would be correct to talk, not about anharmonicity of a surface as a whole, but about anharmonicity of its individual sections. In the present work the ratios ~ (3)/~ (2) have been used as parameters that character• R /~ a(2) and ~ ~' a(a)/~ /~' R ize the extent to which the potential curve deviates from a parabola. Let us call them the normalized cubic and quartic coefficients of the Taylor series, and denote them respectively by *F (R3) and *F (R4) . A very limited number of works are dedicated to the computation of potential energy surfaces of systems having hydrogen bonds. In most of these works use has been made of different functions, which makes difficult the comparison of potential surfaces described by them. For ease of comparison, as first topics of study, we selected ten hydrogen-bonded complexes whose energies were calculated by a common system [20]. An analysis of the results of Gribov et al. [20 ] reveals that the force constant of the OH group decreases monotonically as the strength of the hydrogen bond,
498
determined by the shift of the Poll band, is increased. With the strengthening of the hydrogen bond, the starting as well as normalized cubic and quartic coefficients vary randomly (Fig. 1 ). At the same time, a clear-cut dependence between the normalized coefficients of anharmonicity (*F(R3) and *F (R4);Fig. 2, open circles) is observed. It is interesting that the location of points along the curve expressing this dependence is in no way related to the energy of the hydrogen bond. Thus, for instance, the A VOHshift characterizing the strength of hydrogen bond in the CH3OH-'' I - , CH3OH" • •Cl- and CH3OH- • •NC CH3 systems (points A, B and C in Fig. 2 ) amounts respectively to 252, 364 and 86 cm -1 [20]. A specially conducted analysis [21 ] has shown that not only the hydrogen but also the covalent and multiple bonds are subject to the found dependence. Minyaev [22 ] has also pointed to the resemblance of covalent and symmetric hydrogen bonds in the charged complexes. However, for the dependence established by us this resemblance is of common nature. The dynamic anharmonicity of about 70 bonds computed by more than 30 authors obey this dependence [13, 20, 23-50]. Thus, it is to be admitted that the functional dependence of
(zl
-~
,,
a
••
L
2[-
.I
•
•
•
° o
~3jl
n -~
J
..
r_,i3) ~ r_,12) i ~
t.0
°a'°
°e I
•
1
•
--
I
I
I
ooo
°° I °e
200
I - -
• •
•
I
0
e• 2
21 [
0
°
"° A
I
200~
Fig. I . Potential energy surface parameters of h},drogen bridge vs. hydrogen-bond strength, A V o H (after the data of ref. 20): (a) F(~.! B (mdyn A-l); (b) F(o{)H (mdyn A-l); (c) F~!.B (mdyn ~--2); (3) (2) o --I , (4) o --3 . (4) (3) e --I (d) FH..B/FH..B (A ), (e)FI-I...B (mdynA ), (f) FH...B/FH... B (A ).
499
A-')
2,o,1,,,6,
,
~/~,~/ ]1._+.,, ,
~
t
/J .//
I
°'~f:o
20
C. ~.r
o
,
3
d."
Fig. 2. Values of reduced quartic and cubic coefficientsappearing in potential energyexpressions of different bonds: O, hydrogenbonds; 0, single bonds; X, double bonds; A, triple bonds. +, Analogous coefficientsfor isofrequencypolynomialslisted in Table 1; - - and - - - are the dependences between the enumerated coefficientsobtained on expandinginto Taylor series the Lenard-Johns potential and the Morsefunction after varyingin them RL Dand o~M,respectively. the normalized quartic coefficient of a section of the potential surface of a molecule along the coordinate bond (r) (*F(R4) ) on the analogous cubic coefficient, *F (R3) , remains practically unchanged. However, it is not yet clear why not one but an infinite set of possible values v~ ~ *~ . ' R(4)/.~~ ~'R(3) exists for every bond. For instance, in Fig. 2 the points D, E and F obtained respectively by Gribov et al. [25 ], Smith and Overend [43 ] and Parr et al. [48 ] correspond to one and the same C = O bond of a C02 molecule. In order to u n d e r s t a n d why an infinite set of possible values of *F (4)/*F n(3) exists on the curve of Fig. 2, we conducted the following experiment. The vibrational levels lying in the lower quarter of the potential well (Table 1 ) were calculated for a pure Morse potential of depth i eV and coefficient a - - 1 A - I ; here, the reduced mass was taken equal to 10. Then, by solving the inverse problem we found the polynomial potentials which with reasonable accuracy reproduce the frequencies of the initial Morse function. On varying the ratio of the coefficients of the polynomial between 1: - 2:1.5 and 1: - 7:31 the error in describing the frequencies was found not to exceed 2% (Table 1 ). The points corresponding to the enumerated polynomials are labelled + in Fig. 2. The results obtained convincingly reveal t h a t the following from the litera-
500 TABLE 1 Frequency shifts occuring when Morse function is approximated by Taylor series Exact frequencies of Morse function
Taylor series Ratio of the Taylor series coefficients
163.8 342.2 481.2 634.6 784.7
s
1:2.1:1.5
1:2.5:3
1:3:7
1:4.9:18
1:6.9:31
1.0 2.8 5.3 8.5 12.1
0.4 1.0 1.9 3.1 4.3
0.4 1.4 3.7 6.2 15.1
-3.5 -7.1 -7.3 -1.2 -12.1
-2.8 -5.9 0.0 18.0 48.0
(ev)
/
0.250 --
i~ /
--4
, ,o~ -0.4
-0.2
//.~
L-Y1 , o , ~ 0.2
0.4
0.6
- B e (~)
Fig. 3. Form of some isofrequency potentials for the Taylor series coefficients: 1:2.1:1.5;- • - , 1:6.9:31.
,1:3:7;------,
ture data relationship between the normalized cubic and quartic coefficients of a three-term polynomial function is caused by an almost strict "isofrequency" of the corresponding potentials. In fact, as seen from Fig. 3, the widths of isofrequency potential wells remain almost unchanged. Such wells are distinguished only by the location of their centres relative to the equilibrium position. Thus, searching for a polynomial expression of the form of a potential curve by using the levels lying in its lower quarter is in principle an ill-posed problem. On the other hand, the correct form of the lower part of the crosssection of the potential energy surface can be described (with an accuracy good for calculating the frequencies) by any polynomial series satisfying the found
501
relationship. This feature of the given method of describing the potential energy surface appears to be very important in making specific calculations. The volume of the conducted theoretical studies makes it possible to draw several other practically important conclusions. ( 1 ) The results of concrete calculations [ 28 ] reveal that for a better description of the vibrational spectrum of a molecule it is sufficient to specify the molecule's potential surface in a quasi-diagonal approximation. In so doing, the interaction of coordinates is allowed for only by the quadratic terms. (2) From the results of refs. 13, 21, 23 and 51 it is evident that during hydrogen bridge formation the decrease in the VoH vibration frequency is partly (merely one quarter) caused by the coefficient of interaction of A-H and H-.- B bonds and three quarters by the decrease in the force constant of the A-H bond itself. In the zero approximation, the relationship between all force constants of a hydrogen bridge may be assumed as linear (Fig. 4). (3) For rough estimation of force constants of angles adjacent to the hydrogen bonds use can be made of the relationship F(A~c (mdyn £ ) = 0.0866RAB (£)RBc (£) X
[13,51]
[~(2) z ' AB (mdyn .~.-'). F (B~ (mdyn .~-i )] i/2 (4) Because of the large error with which the quantum-chemical calculations presently enable the quartic coefficients appearing in the potential energy expression to be computed, it is recommended that vibrational spectra be calculated with due regard for the quadratic and cubic terms [17 ]. The mathF 2)(=d~.~-1) 8 0
6--
4
2
0
O.5
1.0
1.5
F(2)
H...O (mdyn.~ -1 )
Fig. 4. Dependence between qua~atic coe~cients that appear in the po~ntial energy expression ~ r hydrogen bridge: (a) -O-H,O-H,~(2)" (b) F ~ , H . . . B .
502
2
1.5
2O 1
Do 0.5
I
0.5
I
I
5 I
I
1.5
2
f
-0.5
0.0
0.5
Fig. 5. Form of the potential energy surface of symmetrical hydrogen bridge: (a) in R x r H and R H .x2 coordinates; (b) section on the S - S plane, Rx,ux2 = 2.65 A (based on ref. 13).
ematical essence of this recommendation can be readily understood if we proceed from the concept of "isofrequency" of the above considered potential functions. (5) Scarce data on the form of sections of the potential energy surface of hydrogen bridges [13, 22] at the X1-H and H-" "Xe bond coordinates suggest that these sections have one minimum. A curve with two minima may appear only in a single-coordinate description of the force field of the X1-H---X2 bridge when the distance between X1 and X2 atoms is more than the equilibrium distance (Fig. 5). Summing up the above account, we must admit that the theoretical works available to date on hydrogen-bonded complexes make it possible to propose a unified form of describing anharmonic dynamic parameters of hydrogen bridges of different strengths. It can be hoped that these parameters will enable the main spectral manifestations of hydrogen-bonded complexes to be described. How much this assumption is justified can be judged by comparing the spectra computed on this assumption with the experimental ones. To this question are devoted the following two sections where, as examples, are considered the systems with strong symmetrical and weak hydrogen bonds. DISOLVATES OF A PROTON
We used the proposed procedure of allowing for dynamic anharmonicity to compute the vibrational spectra of complexes containing strong hydrogen bonds. The following ions of symmetry Ceh: H3 O2, H50 + , (CH3OH) 2H + and
503
(CH~CN) 2H +, differing in charge, structure and properties imparted by their functional groups were studied. Intentionally we restricted ourselves to the consideration of only symmetric charged complexes. The point is that the ( X - - . H - - . X ) -+ fragment has a number of characteristic parameters which remain roughly constant in many of the systems containing this fragment. In fact, the energy released upon adding the second solvating molecule to the proton (as a solvating molecule can act the molecules of water, alcohols, ketones, acids and aldehydes) amounts to 30.8 +_2 kcal mol- 1 [52-54 ]. The length of the X - - . H bond is also less sensitive to the ligand structure [22]. These facts, supported by the results of quantum-chemical calculations, lead to the conclusion that the ( X . - . H . - - X ) -+ fragment can be regarded as an independent functional group which in many ways determines the salient features of the complexes where it (the functional group) is present. This permits us to assume that, besides the earlier enumerated quantities, the dynamic and electro-optical parameters in the considered arrangement also remain constant. Table 2 lists the most interesting results of the calculations of the considered complexes, computations being performed by the programme of Shatokhin et al. [17]. It is seen that the bridge atoms have practically the same charge during solvation of proton with the molecules of water and methyl alcohol. An insignificant decrease in the charge of the central hydrogen atom and an abrupt (twofold) increase in the negative charge of oxygen atoms, observed in the case of Ha O~, are caused by the variation in the total ion charge. In a quite different manner are distributed the charges in a (CH3CN)2H + complex; perhaps this is due to the difference in the electron shells of oxygen and nitrogen atoms. The length of hydrogen bridges in all the complexes was found to be very similar. For all the disolvates of proton the computed force constants of symmetric hydrogen bonds Fx...H are almost equal. The force constants of TABLE 2 Force constants, interatomic distances a n d charge distribution on atoms calculated for X ' . . H ' - - X bridges of four disolvates of p r o t o n Disolvates of proton
XI""H'"X2 F(2) X
b~ (2)
R~..R
Qx
(mdyn£-i)
(mdyn A-')
(A)
(electron (electron charge charge units) units)
2.35 2.00 2.48 2.63
1.76 1.50 1.79 1.70
1.207 1.197 1.203 1.180
-0.840 -0.416 - 0.420 -0.090
.
H~O~ HsO~ (CH3OH)2H+ (CH3CN)2H+
bridge parameters
.
.
.
x, ~.s,~...x2
Q~
0.355 0.411 0.403 0.215
504 coordinate interaction, calculated with lesser accuracy, were corrected in accord with the recommendations of refs. 17 and 51. Because the aim of the present study was to obtain a common qualitative picture of spectra, but not to determine more exact values of vibration frequencies and intensities, the calculations were performed with a relatively small basis set. The principle of constructing the basis sets was the same for all four systems. Besides the harmonic wavefunctions of fundamental vibrations, these sets also contained the wavefunctions of combination vibrations with a total quantum number 2 and of overtones up to 5th order (inclusively). The basis constructed by this method for H30~-, HsO +, (CH3OH)2H + and (CH3CN) 2H + complexes consisted respectively of 77, 176, 547, and 689 functions, which abruptly cut short the calculation time but did not help to attain the variation limit in solving the vibrational problems. The variation in frequencies and intensities as the variation limit is approached was specially studied by considering as an example the spectrum of H50 + ion. The basis dimensions were found to have a weak effect on the qualitative form of the computed spectrum. The error in determining the frequency of fundamental and combination vibrations did not exceed 200 and 500 c m - 1, respectively. The determination of the intensity of every individual band should be regarded merely as a rough estimation. However, the general picture formed by several tens of bands qualitatively correctly reflects the actual state of affairs. In calculating the IR spectra of disolvates of proton, use was made of the dynamic and electro-optical coefficients of the ligands taken from the solution of the corresponding inverse spectral problems [27, 55]. In this work, the parameters of dynamic anharmonicity of all the coordinate bonds have been found in accord with the existing recommendations [28] and the dependence established in ref. 21. Proceeding from the above-stated assumption that the electrooptical parameters of symmetric charged bridge bonds in various compounds are almost constant, the parameters of ( X . ' - H - - o X ) -+ fragments in all the complexes considered were assumed constant and were borrowed from refs. 13 and 51. Figure 6 shows the vibrational spectra of H3 O~-, H5 0~-, (CH3OH)2H + and (CH3CN) 2H + ions in the 4000-0 cm - 1range, which have been calculated with due regard for kinematic, dynamic, and electro-optical anharmonicity of vibrations. Due to lack of information on actual values of these parameters, the halfwidths of absorption bands were taken equal to 100 c m - 1 for constructing these spectra. Exceptions were the cases of very intense bands when, for convenience of representing spectra, the halfwidths were increased up to 200 or 400 cm-1. In this figure the solid lines show the sum spectra, and the dotted lines the individual bands conforming to fundamental oscillations of the ions. The location and intensities of bands of combination vibrations and overtones are shown by straight lines of different thickness. Thin as well as twice or four times thick lines show that the halfwidth of the absorption bands represented
505
6"t0 lcm%0te 50
.
'
50
i
... ::r""=....
5Oo A..-
j' i
A
Fig. 6. Spectra ofdisolvates of proton: (a) [ H - O . . . H - - - O - H ] - ; H
HaC. . . . H'"O-CHa
(b) [ H 2 0 " " H " ' O H 2 ] + ;
(¢)
; (d) [HaC-C-N'-'H"-N-C-CHa] ÷. - - , Sum contour;
,
H bands of fundamental vibrations; ..... experimental curve [56]. by them was taken equal to 100, 200 or 400 cm -1, respectively, for constructing the spectra. From the represented IR spectra of four proton disolvates it is evident that they have several common distinguishing features. Thus, in all the studied cases the frequencies of a large number of combination bands conforming to fundamental vibrations of the ions fall within 3600 and 1000 cm -1. Data on the total number of vibrations having significant intensity are listed in Table 3. In the present calculations, as mentioned earlier, account has been taken only of the combination vibrations with a total quantum n u m b e r 2. The allowance made for the third and fourth order combination vibrations in the case of HsO~ ion (which called for increased dimensions of the basis up to 722 and 2087 functions, respectively) lead to the growth of the number of intense bands (Table 3), but the qualitative picture of the spectrum remained unchanged. Note that all measured absorption spectra of complexes [56 ] are in fair agreem e n t with the computed ones (Fig. 6 (a), (b), (c) dotted lines). This agree-
506 TABLE 3 Combination vibrations of the disolvates of proton in the range 3600-1000 c m - ' Disolvate of proton
Total quantum number
Number of combination vibrations
Number of vibrations with intensity >t
H30~ H~O+ HsO + HsO + (CH~OH)2H + (CH3CN)2H +
2 2 3 4 2 2
32 71 162 259 202 277
17 36 56 85 65 22
(D ~-I) 2
0.05 0.05 0.05 0.05 0.05 0.01
ment strongly supports the idea that the choice of basis sets for the spectral simulation has been done adequately. The bands of combination vibrations and overtones of the spectra of all the considered ions were interpreted. The most intense bands were found to be caused by the interaction of different vibrations of the central proton. The group of numerous bands next in intensity is formed by the bands of combination vibrations of the central proton and the angular coordinates adjacent to the bridge. Comparing the computed spectra (Fig. 6) it is seen that for the H 3 0 ~ , HsO + and (CH3OH)2H + ions the band intensities of the fundamental and combination vibrations are comparable. The spectrum of the (CH3CN)2H + complex differs from those of the enumerated ones by the small absolute intensity of fundamental vibrations and a relatively small fraction of combination vibrations. The latter is probably due to the difficulty of occurrence of resonance between the vibrations of the central proton and the coordinates adjacent to the bridge, the reason being the linear geometry of the complex or the C - N bond multiplicity. Quite a number of calculations were performed which enabled the effect of different dynamic and electro-optical parameters on the general shape of the vibrational spectrum to be estimated. For each of the ions, computations were made with several sets of force constants obtained by using the programme of Shatokhin et al. [17 ] or obtained simultaneously from the computed and literature data. Some variants of the force fields, besides the cubic fields, also included the quartic coefficients in the expansion of the potential energy expression into a Taylor series. The earlier described set of dynamic parameters were recognized as optimal. In all cases the qualitative form of the spectrum remained unchanged upon varying the force field. A special series of calculations has been performed for the H5 O+ ion to reveal how the force constants of interaction of bridge bonds, mechanical and
507 electro-optical anharmonicity of vibrations affect the vibrational spectrum. Consideration of the interaction of X I " " H and H . - . X 2 bonds was found to cause appreciable shift of the absorption bands in the harmonic spectrum. The use of the coefficients of mechanical anharmonicity causes some change in the frequency of vibrations, without affecting the spectral picture as a whole. Significant changes in the nature of the spectrum, which manifest themselves in a large number of intense combination bands in the range 3600-1000 cm-1, is noticed on allowing for the parameters of electro-optical anharmonicity. Here, it is not important whether or not the dynamic anharmonicity of vibrations is simultaneously taken into consideration. The presented results suggest that the considered spectral effect is caused by strong electro-optical anharmonicity of the ( X - - - H - - . X ) -+ fragment. An analysis of the computed spectra of four different proton disolvates reveals that continuous absorption observed in the IR spectra of several systems containing analogous fragments may be caused by a large number of intense bands of combination vibrations in the 3600-1000 c m - 1 range. THE WATERDIMER In this section the proposed method of defining the dynamic and electrooptical anharmonicity of bonds has been employed for computing the spectra of anharmonic vibrations of an open dimer of water. Unlike the above considered complexes, the water dimer represents a system with a usual weak hydrogen bond. The aim of this computation was to find a general contour of the absorption curve, absorption being caused by the transitions between all the interacting levels of this associate. The background absorptions in the frequency ranges 2300-1800 and 1300-700 cm -1 aroused special interest. Absorption in these ranges was reliably recorded in the experiment, but does not find an explanation within the framework of harmonic approximation [10-
121. The vibrational spectrum was calculated by the same programmes [ 16 ] for the following initial data. The geometry of the water dimer (Fig. 7) has been taken from Curtiss and Pople [57]. Therefrom we have borrowed a number of relationships between the parameters of the potential function. The force constants for natural coordinates unaffected by the hydrogen bond (Fq, = Fq4 -- Fq5 , F~6, Fq4~6, Fq4q5) have been taken to be the same as for a free molecule of water [27], and the force constant of hydrogen bond (FQ3)has been found from the experimentally measured constant of centrifugal distortion [58]. The quantities Fq2 , F ~ , Fqlq2 were taken in proportion to Fql, F ~ , and Fq4q~, respectively, with the proportionality coefficients deduced from the results of Murby and Pullin [59]. From this work has been taken the force constant of torsional vibration. The values of force constants of the angles adjacent to the hydrogen bonds have
508
R o .....
q2 ~3"
o
"10
= 2.974
o~~q5°~9
A
O3
Fig. 7. Structure of the water dimer and the internal natural coordinates used. TABLE4 Parameters of the potential energy surface of water dimer (Taylor series approximation) a Initial coordinates x (xy)
q, Q3 q2 ~6 ~9 f17 7 tors. qlq2 q4q5
q~a6
Quadratic coefficients (F(2~ -p(2~ -x xy i
8.450 0.110 7.964 0.697 0.668 0.061 0.060 0.008 - 0.095 -0.101 0.219
Cubic coefficients F ~3)
Quartic coefficients F (4
a
b
a
b
- 41.50 - 0.53 - 39.02 ------
- 46.50 - 0.53 - 43.83 - 0.59 - 0.59 0.006 0.01 --
152.100 1.980 143.350 0.697 0.668 0.061 0.060 0.010
126.00 1.98 118.76 -1.18 -1.18
aAll values are given in mdyne ~ 1 - , , where n is the order of the derivative of energy with respect to the bond coordinates. b e e n e s t i m a t e d b y t h e f o r m u l a s u g g e s t e d i n ref. 13. T h e c u b i c a n d q u a r t i c c o e f ficients of anharmonicity were taken: (a) in proportion to the force constants w i t h c o e f f i c i e n t s - 4 . 9 a n d 18 f o r b o n d s , a n d 0 a n d 1 f o r a n g l e s [ 2 1 ] ; ( b ) f o r most coordinates, the coefficients had the same value as for a free molecule of water [27], for a hydrogen bond the earlier values were taken and the values F ~ 3) a n d F ~ 3) h a v e b e e n b o r r o w e d f r o m ref. 58. A l l t h e v a l u e s o f f o r c e c o n s t a n t s u s e d a r e l i s t e d i n T a b l e 4. T h e e l e c t r o - o p t i c a l p a r a m e t e r s f o r t h e p r e s e n t calculations (Table 5) were chosen in the following manner. The values of d i p o l e m o m e n t s f o r a l l b a n d s h a v e b e e n t a k e n f r o m ref. 58. T h e f i r s t a n d s e c o n d
509 TABLE 5 Electro-optical parameters of water dimer (D, D A Bond
O-H O-H'-" O--'H
4000
~,D~. -2)
Parameter
1.515 1.515 0.465
Ol~ Oq
O# c3q'
Op 0,8
Op Off'
Op c3~
O,u Oa
02,u Oqe
02~ Oq'2
02~ 02~ c3qOq' OqOa
02~ Oq' Oa
02p Oa2
0.68 1.08 0.09
-0.14 0.08 0
0 0 0.55
0 0 0
0 0 0
0.62 0.62 --
-0.26 -0.04 -0.07
-1.25 -0.90
-0.20 -0.20
0.20 0.15
1.80 1.20
ZOO0
1.10 0.75
O'
Fig. 8. A b s o r p t i o n s p e c t r u m of t h e w a t e r d i m e r : - - , vibrations.
s u m c o n t o u r ; - - - , b a n d s of f u n d a m e n t a l
derivatives of dipole moments with respect to natural coordinates have been taken from ref. 60. Figure 8 shows the vibrational spectrum of the water dimer in the 4000-0 c m - 1range, computed with consideration for kinematic, dynamic (second variant, Table 4) and electro-optical anharmonicity of vibrations. Calculations were performed using a basis set of 1349 harmonic oscillatory functions. This made it possible to compute frequencies and intensities not only of the fundamental vibrations (dotted lines in Fig. 8) but also of all overtones up to the 5th order inclusively, and also the combination vibrations having a total quantum number less than or equal to 4. For constructing the spectrum the halfwidth of all the constituent bands was taken equal to 100 c m - 1. From the given spectrum it is seen that in the far-IR region there are four intense bands caused by the torsional and libration vibrations of the dimer of water. In the frequency ranges 2300-1800 and 1300-700 cm -1 the absorption caused by the presence of a large number of overlapping bands of the combination vibrations and overtones is noticed. Their intensity is found to be quite significant; in the 1000-50 c m - 1region these bands amount to 35-40% of total absorption. The computed spectrum contains, besides the above considered bands, two more intense absorption bands. The one lying near 3900 cm-~ is caused by
510 combination vibrations, and the other near 1000 cm -1 by the overtone of libration vibration. The computational accuracy of the location and intensity of individual bands point only to the possible appearance of a distinct band of combination vibration (frequency more than those of stretching vibrations) and a band of overtone of libration vibration (frequency more than that of fundamental intermolecular vibrations) in the water dimer spectrum. This conclusion is in accord with the results of the experiments conducted by the matrix isolation technique [61-64]. The authors of refs. 61-63 have actually noticed a band (3776 c m - 1 ) lying above all the bands of valent vibrations, which in refs. 62 and 63 has been interpreted as a vibration-rotation band. In the 600-500 cm-1 range a solitary band lying far from the bands of other intermolecular vibrations has been noticed by the authors of refs. 63 and 64 who have interpreted it as that of fundamental intermolecular vibrations. Possibly, our results shall cause these interpretations to change. The results of this calculation reveal that for complexes having weak hydrogen bonds the proposed way of allowing for dynamic and electro-optical anharmonicity of molecules enables one to correctly convey the salient features of their spectra which cannot be described by the harmonic model. CONCLUSIONS The possibility of describing the spectra of complexes, having weak and strong hydrogen bonds, within the framework of the unified method of estimating the parameters of dynamic and electro-optical anharmonicity of bonds permits us to think that the established dependences correctly reflect in outline the nature and properties of a hydrogen bond.
REFERENCES 1 G.V.Yukhnevich, Opt. Spektrosk. (Akad. Nauk SSSR, 0td. Fiz.-Mat. Nauk), 2 (1963) 223. 2 G.V. Yukhnevich, Infrakrasnaya Spektroskopiya Vody (Infrared Spectroscopy of Water), Nauka, Moscow, 1973. 3 J.R. Scherer, in P.J. Clark and R.E. Heates (Eds.), Adv. Infrared and Raman Spectroscopy, Vol. 5, Heyden, London, 1978,p. 149. 4 G.V. Yukhnevich and B.P. Shelyukhaev, XVII Vsesoyuznii S'ezd po Spektroskopii, Tezisy Dokladov (The XVII All-Union Congress on Spectroscopy, Abstracts of Reports), Minsk, 1971, p. 252. 5 T.A. Ackermann, Z. Phys. Chem (Frankfurt am Main), 27 (1961) 252. 6 G. Zundel, in P. Schuster, G. Zundel and C. Sandorfy (Eds.), The Hydrogen Bond, North Holland, Amsterdam, 1976, p. 683. 7 N.B.Librovich, V.P. Sakun and N.D. Sokolov,VodorodnayaSvyaz (HydrogenBond), Nauka, Moscow, 1981,p. 174. 8 J.H. McCoy, D.B. Rench and R.K. Long, Appl. Opt., 8 (1969) 1471. 9 K.J. Bignell, Q.J.R. Meteorol. Soc., 96 (1970) 390.
511 10 V.V. Fomin, Molekularnoye Pogloszeniye v Infrakrasnikh Oknakh Prozratchnosti (Molecular Absorption in Infrared Atmospheric Windows), Nauka, Novosibirsk, 1986. 11 A.A. Vigasin, Izv. Akad. Nauk SSSR, 19 (1983) 542. 12 A.A. Vigasin and G.V. Chlenova, Izv. Akad. Nauk SSSR, 20 (1984) 657. 13 G.V. Yukhnevich, E.G. Kokhanova, A.I. Pavlyuchko and V.V. Volkov, J. Mol. Struct., 122 (i985) i. 14 L.A. Gribov, Opt. Spektrosk., 31 (1971) 842. 15 I. Suzuki, Bull. Chem. Soc. Jpn., 44 (1971) 3277. 16 A.I. Pavlyuchko and L.A. Gribov, Zh. Prikl. Spektrosk., 33 (1980) 904 17 S.A. Shatokhin, L.A. Gribov and I.S. Perelygin, Zh. Strukt. Khim., 27 (1986) 22. 18 V.P. Bulychev and N.D. Sokolov, Vodorodnaya Svyaz (Hydrogen Bond), Nauka, Moscow, 1981, p. 10. 19 N.D. Sokolov, Vodorodnaya Svyaz (Hydrogen Bond), Nauka, Moscow, 1981, p. 63. 20 L.A. Gribov, T.T. Merzlyak and I.S. Perelygin, Zh. Prikl. Spektrosk., 35 (1981) 688. 21 G.V. Yukhnevich, Dokl. Akad. Nauk SSSR, 288 {1986) 1386. 22 R.M. Minyaev, Zh. Strukt. Khim., 27 (1986) 130. 23 Y. Bouteiller, M.A. AUavena and J.M. Leclercq, J. Chem. Phys., 73 (1980) 2851. 24 A.C. Jannotte, C. Marcott and J. Overend, J. Chem. Phys., 68 (1978) 2076. 25 L.A. Gribov, A.I. Pavlyuchko and G.F. Lozenko, Zh. Prikl. Spektrosk., 36 (1982) 87,274. 26 L.A. Gribov and G.F. Lozenko, Zh. Prikl Spektrosk., 36 (1982) 976. 27 A.I. Pavlyuchko, G.F. Lozenko, G.V. Khovrin and L.A. Gribov, Opt. Spektrosk., 52 (1982) 64. 28 A.I. Pavlyuchko and L.A. Gribov, Opt. Spektrosk., 54 (1983) 644. 29 V.V. Rossikhin and V.P. Morozov. Potentsialniye i Elektroopticheskiye Postoyanniye Molekul (Potential and Electrooptical Constants of Molecules), Energoatomizdat, Moscow, 1983. 30 A.B. Anderson and R.G. Parr, J. Chem. Phys., 53 (1970) 3375; 58 (1973) 381. 31 A. Barbe, C. Secrown and P. Jouve, J. Mol. Spectrosc, 49 (1974) 171. 32 P. Botschwina, W. Meyer and A. Semkov, Chem. Phys., 37 (1976) 25. 33 P. Botschwina, Chem. Phys., 40 (1979) 33. 34 P. Botschwina, H. Haevthner and W. Sawadny, Chem. Phys. Lett., 74 (1980) 156. 35 Z.I. Hardwick and Z.C.D. Brand, Can. J. Phys., 54 (1976) 80. 36 A.R. Hoy, I.M. Mills and G. Strey, J. Mol. Phys., 6 (1972) 1265. 37 K. Machida, J. Chem. Phys., 44 (1966) 4186. 38 K. Machida and Y. Tanaka, J. Chem. Phys., 61 (1974) 5040. 39 I.M. Mills, Theor. Chem., Quantum Chem., 1 (1974) 110. 40 J. Pliva, Proc. Natl. Acad. Sci. U.S.A., 5 (1974) 283. 41 A. Schmiedekamp, C.W. Bock and P. George, J. Mol. Struct., 67 (1980) 107. 42 H.B. Shlegel, S. Wolfe and F. Bernardi, Can. J. Chem., 53 (1975) 3599; J. Chem. Phys., 63 (1975) 3632. 43 D.F. Smith and J. Overend, J. Chem. Phys., 55 (1971) 1157. 44 G. Strey and I.M. Mills, Mol. Phys., 26 (1973) 129. 45 G. Strey and I.M. Mills, J. Mol. Spectrosc., 59 (1976) 103. 46 I. Suzuki, J. Mol. Spectrosc., 25 (1968) 479. 47 Y. Tanaka and K. Machida, J. Mol. Spectrosc., 64 (1977) 429. 48 R.G. Parr, H.F. Hameka and H. Kim, J. Chem. Phys., 49 (1968) 3071, 4849. 49 V. Wahlgren, J. Pacansky and P.S. Bagus, J. Chem. Phys., 63 (1975) 2874. 50 V.K. Wang and J. Overend, Spectrochim. Acta, Part A, 32 (1976) 1043. 51 G.V. Yukhnevich and E.G. Kokhanova, Zh. Prikl. Spektrosk., 39 (1983) 617. 52 Y.K. Lau and P.P.S. Saluja, J. Am. Chem. Soc., 102 (1980) 7429. 53 D.S. Bomse and J.L. Beauchamp, J. Phys. Chem., 85 (1981) 488. 54 J.W. Larson and T.B. McMahon, J. Am. Chem. Soc., 104 (1982) 6255.
512 55 56
57 58 59 60 61 62 63 64
L.A. Gribov, Vvedeniye v Molekulyarnuyu Spektroskopiyu (Introduction to Molecular Spectroscopy), Nauka, Moscow, 1976. E.G. Tarakanova, G.V. Yukhnevich, V.D. Mayorov and N.B. Librovich, Vsesoyuznoye Soveszaniye Kislotno-osnovniye Vzaimodeistviya i Solvatatsiya v Nevodnikh Rastvorakh, Tezisy Dokladov (The All-Union Conference on Acid-Base Interactions and Solvation in Nonwater Solutions, Abstracts of Reports), Kharkov, 1987, p. 145. L.A. Curtiss and J.A. Pople, J. Mol. Spectrosc., 55 (1975) 1. J.A. Odutola and T.R. Dyke, J. Chem Phys., 72 (1980) 5062. E.J. Murby and A.D.E. Pullin, Aust. J. Chem., 32 (1979) 1167. B.P. Shelyukhaev and G.V. Yukhnevich, Opt. Spektrosk., 41 (1976) 404. E.D. Radchenko, G.G. Sheina and Yu.P. Blagoi, Stud. Biophys., 111 (1986) 115. K.B. Harvey, H.F. Shurvell, J. Mol. Spectrosc., 25 (1968) 120. R.M. Bentwood, A.J. Barnes, W.J. Orville-Thomas, J. Mol. Spectrosc., 84 (1980) 391. M.J. Wdjcik, S.A. Rice, J. Chem. Phys., 84 (1986) 3042.
495
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
S O M E P R O P E R T I E S OF T H E P O T E N T I A L E N E R G Y S U R F A C E A N D V I B R A T I O N A L S P E C T R U M OF A S T R O N G H Y D R O G E N BOND COMPLEX
G.V. YUKHNEVICH and E.G. TARAKANOVA
N.S. Kurnakov Institute of General and Inorganic Chemistry, Academy of Sciences of the U.S.S.R., 31 Leninsky Avenue, Moscow 117071 (U.S.S.R.) (Received 14 September 1987)
ABSTRACT The forms of potential energy surfaces of weak and strong hydrogen bonds are analyzed. A method is proposed to take account of anharmonicity of vibrations of A - H - - . B hydrogen bridges. With this method are calculated the spectra of water dimer and of four complexes having symmetric hydrogen bonds. Good agreement between the computed and experimentally obtained spectra suggests that the proposed method can be adequately applied to compute vibrational spectra of systems containing hydrogen bonds.
INTRODUCTION
Since its postulation in the 1930s the hydrogen bond has been regarded as a purely specific form of interaction. Perhaps this is the reason why all its spectral manifestations have been explained by its individuality. With these manifestations were primarily placed the red shift of the stretching vibrations band of the O-H bond (VOH) and the weaker blue shift of the band of its bending vibrations (v~). Also, the gain in the intensities of VOHbands and their mutual location in the isotope-substituted molecules and in different OHx groups were thought to be typical exclusively for the hydrogen bond. The nature of most of the enumerated effects was understood in the 1960s when the semi-empirical theory of vibrations ofpolyatomic molecules was systematically applied to the complexes containing hydrogen bonds. All these effects were found to result from the phenomena well-known by that time in the spectroscopy of organic compounds. It was ascertained [ 1-3 ] that the spectral manifestations of a hydrogen bond were in principle the consequence of variations in the kinematics of the system and of the mutual effect of the neighbouring bonds. The more so, the assumption that a simple linear dependence exists between the dynamic parameters of the hydrogen bond made it possible to describe most of its then known spectral properties [2]. 0022-2860/88/$03.50
© 1988 Elsevier Science Publishers B.V.
496 Later a similar situation was shown to prevail in describing the intensities of absorption bands related to the vibrations of the hydrogen bridge [4 ]. Thus, in the 1960-1970s the whole body of theoretical and experimental data suggested that no additional or special assumptions were needed to describe the spectra of systems containing hydrogen bonds. Further development of spectral equipment made it possible to measure spectra in greater detail and to significantly widen the range of the systems studied. As a result, new spectral features of the systems containing hydrogen bonds were noticed, which could not be taken into account from the standpoint of the theory of harmonic vibrations. Thus, in the spectra of many hydrogenbonded liquid and amorphous systems continuous absorption caused by the vibrations of the X l " " - H . . "X2 fragment was noticed in the 3600-1000 cm -1 range [ 5, 6 ]. A detailed theoretical analysis of this phenomenon has revealed that continuous absorption cannot be caused by the medium's effect [7]. Slightly later, absorption in the spectra of water vapour was recorded in the 1200-800 cm-1 range where there can be none of the fundamental vibrations of water clusters [8, 9]. In the past few years, this absorption has frequently been related to the water dimers [10-12 ]. In both the examples the observed absorption is ascribed to the combination bands caused by vibrations including the hydrogen bond [ 12, 13 ]. As a check on this assumption it was necessary to quantitatively characterize the anharmonicity of hydrogen bonds of different strength. OBJECT OF INVESTIGATION In this paper an attempt has been made to provide a common system for describing the potential energy surface of complexes having hydrogen bonds. The coefficients of dynamic anharmonicity found by the proposed method were used to calculate the vibrational spectra of a number of systems having strong and weak hydrogen bonds. The effect of electro-optical and mechanical anharmonicity of vibrations on the nature of computed spectra has been analyzed. The calculated spectra have been compared with the experimental ones. This work was made possible by considerable progress made in quantum chemistry, the theory of anharmonic vibrations and computer facilities. The bases of the method of computing anharmonic vibrations in polyatomic molecules were laid in refs. 14 and 15. Therein it was proposed to perform direct numerical diagonalization of the vibrational part of the Hamiltonian using Ritz's variation method. The vibrational spectra considered below were computed by a set of programmes [ 16 ] where the basis set contained up to 10 000 harmonic oscillatory functions. Equilibrium configurations, force constants and the distribution of charges on the atoms of the studied molecules were found by using a set of programmes [ 17 ].
497 When this work was undertaken, information on the form of the potential surface of hydrogen bond was very scanty. The quantum-chemical studies were persued in a somewhat different direction. Much attention was paid to checking the correctness of equilibrium configurations computed for complexes having strong and weak hydrogen bonds [18]. In studying different interaction mechanisms it was shown that the donor-acceptor interaction played an insignificant role in the case of intermolecular and ion-molecular hydrogen bonds. The main contribution was made by the Coulomb and exchange forces [18]. Also, the potential curves of strong and weak hydrogen bonds were considered to have a completely different form. It was assumed that strong asymmetric hydrogen bonds had two minima and the decrease in the VOH vibration frequency was due to the dynamic interaction of A-H and H-- .B bonds of the bridge [ 19 ]. All this was in accord with the intuitive concept of the specific nature of a hydrogen bond. SOME DYNAMICCHARACTERISTICSOF A HYDROGENBOND Before we embark on the study of dynamic anharmonicity of hydrogen bonds, it is necessary to choose an analytical form for describing the potential energy surface and the parameters that would definitely reflect the extent to which the function deviates from the purely harmonic function and would be independent of the bond length and energy. The analytical expressions for the form of potential energy surface of a molecule may be diverse. In the literature, the form is more often described by a Taylor polynomial series, usually containing quadratic, cubic and quartic terms with coefficients F (R 2) , F (R 3) and F (R 4) , respectively. To facilitate working with the literature data, we shall also make use of this form of the expression. Obviously, the form of the potential energy surface of a hydrogen bridge, like any multidimensional surface, cannot be described by one parameter. Therefore, it would be correct to talk, not about anharmonicity of a surface as a whole, but about anharmonicity of its individual sections. In the present work the ratios ~ (3)/~ (2) have been used as parameters that character• R /~ a(2) and ~ ~' a(a)/~ /~' R ize the extent to which the potential curve deviates from a parabola. Let us call them the normalized cubic and quartic coefficients of the Taylor series, and denote them respectively by *F (R3) and *F (R4) . A very limited number of works are dedicated to the computation of potential energy surfaces of systems having hydrogen bonds. In most of these works use has been made of different functions, which makes difficult the comparison of potential surfaces described by them. For ease of comparison, as first topics of study, we selected ten hydrogen-bonded complexes whose energies were calculated by a common system [20]. An analysis of the results of Gribov et al. [20 ] reveals that the force constant of the OH group decreases monotonically as the strength of the hydrogen bond,
498
determined by the shift of the Poll band, is increased. With the strengthening of the hydrogen bond, the starting as well as normalized cubic and quartic coefficients vary randomly (Fig. 1 ). At the same time, a clear-cut dependence between the normalized coefficients of anharmonicity (*F(R3) and *F (R4);Fig. 2, open circles) is observed. It is interesting that the location of points along the curve expressing this dependence is in no way related to the energy of the hydrogen bond. Thus, for instance, the A VOHshift characterizing the strength of hydrogen bond in the CH3OH-'' I - , CH3OH" • •Cl- and CH3OH- • •NC CH3 systems (points A, B and C in Fig. 2 ) amounts respectively to 252, 364 and 86 cm -1 [20]. A specially conducted analysis [21 ] has shown that not only the hydrogen but also the covalent and multiple bonds are subject to the found dependence. Minyaev [22 ] has also pointed to the resemblance of covalent and symmetric hydrogen bonds in the charged complexes. However, for the dependence established by us this resemblance is of common nature. The dynamic anharmonicity of about 70 bonds computed by more than 30 authors obey this dependence [13, 20, 23-50]. Thus, it is to be admitted that the functional dependence of
(zl
-~
,,
a
••
L
2[-
.I
•
•
•
° o
~3jl
n -~
J
..
r_,i3) ~ r_,12) i ~
t.0
°a'°
°e I
•
1
•
--
I
I
I
ooo
°° I °e
200
I - -
• •
•
I
0
e• 2
21 [
0
°
"° A
I
200~
Fig. I . Potential energy surface parameters of h},drogen bridge vs. hydrogen-bond strength, A V o H (after the data of ref. 20): (a) F(~.! B (mdyn A-l); (b) F(o{)H (mdyn A-l); (c) F~!.B (mdyn ~--2); (3) (2) o --I , (4) o --3 . (4) (3) e --I (d) FH..B/FH..B (A ), (e)FI-I...B (mdynA ), (f) FH...B/FH... B (A ).
499
A-')
2,o,1,,,6,
,
~/~,~/ ]1._+.,, ,
~
t
/J .//
I
°'~f:o
20
C. ~.r
o
,
3
d."
Fig. 2. Values of reduced quartic and cubic coefficientsappearing in potential energyexpressions of different bonds: O, hydrogenbonds; 0, single bonds; X, double bonds; A, triple bonds. +, Analogous coefficientsfor isofrequencypolynomialslisted in Table 1; - - and - - - are the dependences between the enumerated coefficientsobtained on expandinginto Taylor series the Lenard-Johns potential and the Morsefunction after varyingin them RL Dand o~M,respectively. the normalized quartic coefficient of a section of the potential surface of a molecule along the coordinate bond (r) (*F(R4) ) on the analogous cubic coefficient, *F (R3) , remains practically unchanged. However, it is not yet clear why not one but an infinite set of possible values v~ ~ *~ . ' R(4)/.~~ ~'R(3) exists for every bond. For instance, in Fig. 2 the points D, E and F obtained respectively by Gribov et al. [25 ], Smith and Overend [43 ] and Parr et al. [48 ] correspond to one and the same C = O bond of a C02 molecule. In order to u n d e r s t a n d why an infinite set of possible values of *F (4)/*F n(3) exists on the curve of Fig. 2, we conducted the following experiment. The vibrational levels lying in the lower quarter of the potential well (Table 1 ) were calculated for a pure Morse potential of depth i eV and coefficient a - - 1 A - I ; here, the reduced mass was taken equal to 10. Then, by solving the inverse problem we found the polynomial potentials which with reasonable accuracy reproduce the frequencies of the initial Morse function. On varying the ratio of the coefficients of the polynomial between 1: - 2:1.5 and 1: - 7:31 the error in describing the frequencies was found not to exceed 2% (Table 1 ). The points corresponding to the enumerated polynomials are labelled + in Fig. 2. The results obtained convincingly reveal t h a t the following from the litera-
500 TABLE 1 Frequency shifts occuring when Morse function is approximated by Taylor series Exact frequencies of Morse function
Taylor series Ratio of the Taylor series coefficients
163.8 342.2 481.2 634.6 784.7
s
1:2.1:1.5
1:2.5:3
1:3:7
1:4.9:18
1:6.9:31
1.0 2.8 5.3 8.5 12.1
0.4 1.0 1.9 3.1 4.3
0.4 1.4 3.7 6.2 15.1
-3.5 -7.1 -7.3 -1.2 -12.1
-2.8 -5.9 0.0 18.0 48.0
(ev)
/
0.250 --
i~ /
--4
, ,o~ -0.4
-0.2
//.~
L-Y1 , o , ~ 0.2
0.4
0.6
- B e (~)
Fig. 3. Form of some isofrequency potentials for the Taylor series coefficients: 1:2.1:1.5;- • - , 1:6.9:31.
,1:3:7;------,
ture data relationship between the normalized cubic and quartic coefficients of a three-term polynomial function is caused by an almost strict "isofrequency" of the corresponding potentials. In fact, as seen from Fig. 3, the widths of isofrequency potential wells remain almost unchanged. Such wells are distinguished only by the location of their centres relative to the equilibrium position. Thus, searching for a polynomial expression of the form of a potential curve by using the levels lying in its lower quarter is in principle an ill-posed problem. On the other hand, the correct form of the lower part of the crosssection of the potential energy surface can be described (with an accuracy good for calculating the frequencies) by any polynomial series satisfying the found
501
relationship. This feature of the given method of describing the potential energy surface appears to be very important in making specific calculations. The volume of the conducted theoretical studies makes it possible to draw several other practically important conclusions. ( 1 ) The results of concrete calculations [ 28 ] reveal that for a better description of the vibrational spectrum of a molecule it is sufficient to specify the molecule's potential surface in a quasi-diagonal approximation. In so doing, the interaction of coordinates is allowed for only by the quadratic terms. (2) From the results of refs. 13, 21, 23 and 51 it is evident that during hydrogen bridge formation the decrease in the VoH vibration frequency is partly (merely one quarter) caused by the coefficient of interaction of A-H and H-.- B bonds and three quarters by the decrease in the force constant of the A-H bond itself. In the zero approximation, the relationship between all force constants of a hydrogen bridge may be assumed as linear (Fig. 4). (3) For rough estimation of force constants of angles adjacent to the hydrogen bonds use can be made of the relationship F(A~c (mdyn £ ) = 0.0866RAB (£)RBc (£) X
[13,51]
[~(2) z ' AB (mdyn .~.-'). F (B~ (mdyn .~-i )] i/2 (4) Because of the large error with which the quantum-chemical calculations presently enable the quartic coefficients appearing in the potential energy expression to be computed, it is recommended that vibrational spectra be calculated with due regard for the quadratic and cubic terms [17 ]. The mathF 2)(=d~.~-1) 8 0
6--
4
2
0
O.5
1.0
1.5
F(2)
H...O (mdyn.~ -1 )
Fig. 4. Dependence between qua~atic coe~cients that appear in the po~ntial energy expression ~ r hydrogen bridge: (a) -O-H,O-H,~(2)" (b) F ~ , H . . . B .
502
2
1.5
2O 1
Do 0.5
I
0.5
I
I
5 I
I
1.5
2
f
-0.5
0.0
0.5
Fig. 5. Form of the potential energy surface of symmetrical hydrogen bridge: (a) in R x r H and R H .x2 coordinates; (b) section on the S - S plane, Rx,ux2 = 2.65 A (based on ref. 13).
ematical essence of this recommendation can be readily understood if we proceed from the concept of "isofrequency" of the above considered potential functions. (5) Scarce data on the form of sections of the potential energy surface of hydrogen bridges [13, 22] at the X1-H and H-" "Xe bond coordinates suggest that these sections have one minimum. A curve with two minima may appear only in a single-coordinate description of the force field of the X1-H---X2 bridge when the distance between X1 and X2 atoms is more than the equilibrium distance (Fig. 5). Summing up the above account, we must admit that the theoretical works available to date on hydrogen-bonded complexes make it possible to propose a unified form of describing anharmonic dynamic parameters of hydrogen bridges of different strengths. It can be hoped that these parameters will enable the main spectral manifestations of hydrogen-bonded complexes to be described. How much this assumption is justified can be judged by comparing the spectra computed on this assumption with the experimental ones. To this question are devoted the following two sections where, as examples, are considered the systems with strong symmetrical and weak hydrogen bonds. DISOLVATES OF A PROTON
We used the proposed procedure of allowing for dynamic anharmonicity to compute the vibrational spectra of complexes containing strong hydrogen bonds. The following ions of symmetry Ceh: H3 O2, H50 + , (CH3OH) 2H + and
503
(CH~CN) 2H +, differing in charge, structure and properties imparted by their functional groups were studied. Intentionally we restricted ourselves to the consideration of only symmetric charged complexes. The point is that the ( X - - . H - - . X ) -+ fragment has a number of characteristic parameters which remain roughly constant in many of the systems containing this fragment. In fact, the energy released upon adding the second solvating molecule to the proton (as a solvating molecule can act the molecules of water, alcohols, ketones, acids and aldehydes) amounts to 30.8 +_2 kcal mol- 1 [52-54 ]. The length of the X - - . H bond is also less sensitive to the ligand structure [22]. These facts, supported by the results of quantum-chemical calculations, lead to the conclusion that the ( X . - . H . - - X ) -+ fragment can be regarded as an independent functional group which in many ways determines the salient features of the complexes where it (the functional group) is present. This permits us to assume that, besides the earlier enumerated quantities, the dynamic and electro-optical parameters in the considered arrangement also remain constant. Table 2 lists the most interesting results of the calculations of the considered complexes, computations being performed by the programme of Shatokhin et al. [17]. It is seen that the bridge atoms have practically the same charge during solvation of proton with the molecules of water and methyl alcohol. An insignificant decrease in the charge of the central hydrogen atom and an abrupt (twofold) increase in the negative charge of oxygen atoms, observed in the case of Ha O~, are caused by the variation in the total ion charge. In a quite different manner are distributed the charges in a (CH3CN)2H + complex; perhaps this is due to the difference in the electron shells of oxygen and nitrogen atoms. The length of hydrogen bridges in all the complexes was found to be very similar. For all the disolvates of proton the computed force constants of symmetric hydrogen bonds Fx...H are almost equal. The force constants of TABLE 2 Force constants, interatomic distances a n d charge distribution on atoms calculated for X ' . . H ' - - X bridges of four disolvates of p r o t o n Disolvates of proton
XI""H'"X2 F(2) X
b~ (2)
R~..R
Qx
(mdyn£-i)
(mdyn A-')
(A)
(electron (electron charge charge units) units)
2.35 2.00 2.48 2.63
1.76 1.50 1.79 1.70
1.207 1.197 1.203 1.180
-0.840 -0.416 - 0.420 -0.090
.
H~O~ HsO~ (CH3OH)2H+ (CH3CN)2H+
bridge parameters
.
.
.
x, ~.s,~...x2
Q~
0.355 0.411 0.403 0.215
504 coordinate interaction, calculated with lesser accuracy, were corrected in accord with the recommendations of refs. 17 and 51. Because the aim of the present study was to obtain a common qualitative picture of spectra, but not to determine more exact values of vibration frequencies and intensities, the calculations were performed with a relatively small basis set. The principle of constructing the basis sets was the same for all four systems. Besides the harmonic wavefunctions of fundamental vibrations, these sets also contained the wavefunctions of combination vibrations with a total quantum number 2 and of overtones up to 5th order (inclusively). The basis constructed by this method for H30~-, HsO +, (CH3OH)2H + and (CH3CN) 2H + complexes consisted respectively of 77, 176, 547, and 689 functions, which abruptly cut short the calculation time but did not help to attain the variation limit in solving the vibrational problems. The variation in frequencies and intensities as the variation limit is approached was specially studied by considering as an example the spectrum of H50 + ion. The basis dimensions were found to have a weak effect on the qualitative form of the computed spectrum. The error in determining the frequency of fundamental and combination vibrations did not exceed 200 and 500 c m - 1, respectively. The determination of the intensity of every individual band should be regarded merely as a rough estimation. However, the general picture formed by several tens of bands qualitatively correctly reflects the actual state of affairs. In calculating the IR spectra of disolvates of proton, use was made of the dynamic and electro-optical coefficients of the ligands taken from the solution of the corresponding inverse spectral problems [27, 55]. In this work, the parameters of dynamic anharmonicity of all the coordinate bonds have been found in accord with the existing recommendations [28] and the dependence established in ref. 21. Proceeding from the above-stated assumption that the electrooptical parameters of symmetric charged bridge bonds in various compounds are almost constant, the parameters of ( X . ' - H - - o X ) -+ fragments in all the complexes considered were assumed constant and were borrowed from refs. 13 and 51. Figure 6 shows the vibrational spectra of H3 O~-, H5 0~-, (CH3OH)2H + and (CH3CN) 2H + ions in the 4000-0 cm - 1range, which have been calculated with due regard for kinematic, dynamic, and electro-optical anharmonicity of vibrations. Due to lack of information on actual values of these parameters, the halfwidths of absorption bands were taken equal to 100 c m - 1 for constructing these spectra. Exceptions were the cases of very intense bands when, for convenience of representing spectra, the halfwidths were increased up to 200 or 400 cm-1. In this figure the solid lines show the sum spectra, and the dotted lines the individual bands conforming to fundamental oscillations of the ions. The location and intensities of bands of combination vibrations and overtones are shown by straight lines of different thickness. Thin as well as twice or four times thick lines show that the halfwidth of the absorption bands represented
505
6"t0 lcm%0te 50
.
'
50
i
... ::r""=....
5Oo A..-
j' i
A
Fig. 6. Spectra ofdisolvates of proton: (a) [ H - O . . . H - - - O - H ] - ; H
HaC. . . . H'"O-CHa
(b) [ H 2 0 " " H " ' O H 2 ] + ;
(¢)
; (d) [HaC-C-N'-'H"-N-C-CHa] ÷. - - , Sum contour;
,
H bands of fundamental vibrations; ..... experimental curve [56]. by them was taken equal to 100, 200 or 400 cm -1, respectively, for constructing the spectra. From the represented IR spectra of four proton disolvates it is evident that they have several common distinguishing features. Thus, in all the studied cases the frequencies of a large number of combination bands conforming to fundamental vibrations of the ions fall within 3600 and 1000 cm -1. Data on the total number of vibrations having significant intensity are listed in Table 3. In the present calculations, as mentioned earlier, account has been taken only of the combination vibrations with a total quantum n u m b e r 2. The allowance made for the third and fourth order combination vibrations in the case of HsO~ ion (which called for increased dimensions of the basis up to 722 and 2087 functions, respectively) lead to the growth of the number of intense bands (Table 3), but the qualitative picture of the spectrum remained unchanged. Note that all measured absorption spectra of complexes [56 ] are in fair agreem e n t with the computed ones (Fig. 6 (a), (b), (c) dotted lines). This agree-
506 TABLE 3 Combination vibrations of the disolvates of proton in the range 3600-1000 c m - ' Disolvate of proton
Total quantum number
Number of combination vibrations
Number of vibrations with intensity >t
H30~ H~O+ HsO + HsO + (CH~OH)2H + (CH3CN)2H +
2 2 3 4 2 2
32 71 162 259 202 277
17 36 56 85 65 22
(D ~-I) 2
0.05 0.05 0.05 0.05 0.05 0.01
ment strongly supports the idea that the choice of basis sets for the spectral simulation has been done adequately. The bands of combination vibrations and overtones of the spectra of all the considered ions were interpreted. The most intense bands were found to be caused by the interaction of different vibrations of the central proton. The group of numerous bands next in intensity is formed by the bands of combination vibrations of the central proton and the angular coordinates adjacent to the bridge. Comparing the computed spectra (Fig. 6) it is seen that for the H 3 0 ~ , HsO + and (CH3OH)2H + ions the band intensities of the fundamental and combination vibrations are comparable. The spectrum of the (CH3CN)2H + complex differs from those of the enumerated ones by the small absolute intensity of fundamental vibrations and a relatively small fraction of combination vibrations. The latter is probably due to the difficulty of occurrence of resonance between the vibrations of the central proton and the coordinates adjacent to the bridge, the reason being the linear geometry of the complex or the C - N bond multiplicity. Quite a number of calculations were performed which enabled the effect of different dynamic and electro-optical parameters on the general shape of the vibrational spectrum to be estimated. For each of the ions, computations were made with several sets of force constants obtained by using the programme of Shatokhin et al. [17 ] or obtained simultaneously from the computed and literature data. Some variants of the force fields, besides the cubic fields, also included the quartic coefficients in the expansion of the potential energy expression into a Taylor series. The earlier described set of dynamic parameters were recognized as optimal. In all cases the qualitative form of the spectrum remained unchanged upon varying the force field. A special series of calculations has been performed for the H5 O+ ion to reveal how the force constants of interaction of bridge bonds, mechanical and
507 electro-optical anharmonicity of vibrations affect the vibrational spectrum. Consideration of the interaction of X I " " H and H . - . X 2 bonds was found to cause appreciable shift of the absorption bands in the harmonic spectrum. The use of the coefficients of mechanical anharmonicity causes some change in the frequency of vibrations, without affecting the spectral picture as a whole. Significant changes in the nature of the spectrum, which manifest themselves in a large number of intense combination bands in the range 3600-1000 cm-1, is noticed on allowing for the parameters of electro-optical anharmonicity. Here, it is not important whether or not the dynamic anharmonicity of vibrations is simultaneously taken into consideration. The presented results suggest that the considered spectral effect is caused by strong electro-optical anharmonicity of the ( X - - - H - - . X ) -+ fragment. An analysis of the computed spectra of four different proton disolvates reveals that continuous absorption observed in the IR spectra of several systems containing analogous fragments may be caused by a large number of intense bands of combination vibrations in the 3600-1000 c m - 1 range. THE WATERDIMER In this section the proposed method of defining the dynamic and electrooptical anharmonicity of bonds has been employed for computing the spectra of anharmonic vibrations of an open dimer of water. Unlike the above considered complexes, the water dimer represents a system with a usual weak hydrogen bond. The aim of this computation was to find a general contour of the absorption curve, absorption being caused by the transitions between all the interacting levels of this associate. The background absorptions in the frequency ranges 2300-1800 and 1300-700 cm -1 aroused special interest. Absorption in these ranges was reliably recorded in the experiment, but does not find an explanation within the framework of harmonic approximation [10-
121. The vibrational spectrum was calculated by the same programmes [ 16 ] for the following initial data. The geometry of the water dimer (Fig. 7) has been taken from Curtiss and Pople [57]. Therefrom we have borrowed a number of relationships between the parameters of the potential function. The force constants for natural coordinates unaffected by the hydrogen bond (Fq, = Fq4 -- Fq5 , F~6, Fq4~6, Fq4q5) have been taken to be the same as for a free molecule of water [27], and the force constant of hydrogen bond (FQ3)has been found from the experimentally measured constant of centrifugal distortion [58]. The quantities Fq2 , F ~ , Fqlq2 were taken in proportion to Fql, F ~ , and Fq4q~, respectively, with the proportionality coefficients deduced from the results of Murby and Pullin [59]. From this work has been taken the force constant of torsional vibration. The values of force constants of the angles adjacent to the hydrogen bonds have
508
R o .....
q2 ~3"
o
"10
= 2.974
o~~q5°~9
A
O3
Fig. 7. Structure of the water dimer and the internal natural coordinates used. TABLE4 Parameters of the potential energy surface of water dimer (Taylor series approximation) a Initial coordinates x (xy)
q, Q3 q2 ~6 ~9 f17 7 tors. qlq2 q4q5
q~a6
Quadratic coefficients (F(2~ -p(2~ -x xy i
8.450 0.110 7.964 0.697 0.668 0.061 0.060 0.008 - 0.095 -0.101 0.219
Cubic coefficients F ~3)
Quartic coefficients F (4
a
b
a
b
- 41.50 - 0.53 - 39.02 ------
- 46.50 - 0.53 - 43.83 - 0.59 - 0.59 0.006 0.01 --
152.100 1.980 143.350 0.697 0.668 0.061 0.060 0.010
126.00 1.98 118.76 -1.18 -1.18
aAll values are given in mdyne ~ 1 - , , where n is the order of the derivative of energy with respect to the bond coordinates. b e e n e s t i m a t e d b y t h e f o r m u l a s u g g e s t e d i n ref. 13. T h e c u b i c a n d q u a r t i c c o e f ficients of anharmonicity were taken: (a) in proportion to the force constants w i t h c o e f f i c i e n t s - 4 . 9 a n d 18 f o r b o n d s , a n d 0 a n d 1 f o r a n g l e s [ 2 1 ] ; ( b ) f o r most coordinates, the coefficients had the same value as for a free molecule of water [27], for a hydrogen bond the earlier values were taken and the values F ~ 3) a n d F ~ 3) h a v e b e e n b o r r o w e d f r o m ref. 58. A l l t h e v a l u e s o f f o r c e c o n s t a n t s u s e d a r e l i s t e d i n T a b l e 4. T h e e l e c t r o - o p t i c a l p a r a m e t e r s f o r t h e p r e s e n t calculations (Table 5) were chosen in the following manner. The values of d i p o l e m o m e n t s f o r a l l b a n d s h a v e b e e n t a k e n f r o m ref. 58. T h e f i r s t a n d s e c o n d
509 TABLE 5 Electro-optical parameters of water dimer (D, D A Bond
O-H O-H'-" O--'H
4000
~,D~. -2)
Parameter
1.515 1.515 0.465
Ol~ Oq
O# c3q'
Op 0,8
Op Off'
Op c3~
O,u Oa
02,u Oqe
02~ Oq'2
02~ 02~ c3qOq' OqOa
02~ Oq' Oa
02p Oa2
0.68 1.08 0.09
-0.14 0.08 0
0 0 0.55
0 0 0
0 0 0
0.62 0.62 --
-0.26 -0.04 -0.07
-1.25 -0.90
-0.20 -0.20
0.20 0.15
1.80 1.20
ZOO0
1.10 0.75
O'
Fig. 8. A b s o r p t i o n s p e c t r u m of t h e w a t e r d i m e r : - - , vibrations.
s u m c o n t o u r ; - - - , b a n d s of f u n d a m e n t a l
derivatives of dipole moments with respect to natural coordinates have been taken from ref. 60. Figure 8 shows the vibrational spectrum of the water dimer in the 4000-0 c m - 1range, computed with consideration for kinematic, dynamic (second variant, Table 4) and electro-optical anharmonicity of vibrations. Calculations were performed using a basis set of 1349 harmonic oscillatory functions. This made it possible to compute frequencies and intensities not only of the fundamental vibrations (dotted lines in Fig. 8) but also of all overtones up to the 5th order inclusively, and also the combination vibrations having a total quantum number less than or equal to 4. For constructing the spectrum the halfwidth of all the constituent bands was taken equal to 100 c m - 1. From the given spectrum it is seen that in the far-IR region there are four intense bands caused by the torsional and libration vibrations of the dimer of water. In the frequency ranges 2300-1800 and 1300-700 cm -1 the absorption caused by the presence of a large number of overlapping bands of the combination vibrations and overtones is noticed. Their intensity is found to be quite significant; in the 1000-50 c m - 1region these bands amount to 35-40% of total absorption. The computed spectrum contains, besides the above considered bands, two more intense absorption bands. The one lying near 3900 cm-~ is caused by
510 combination vibrations, and the other near 1000 cm -1 by the overtone of libration vibration. The computational accuracy of the location and intensity of individual bands point only to the possible appearance of a distinct band of combination vibration (frequency more than those of stretching vibrations) and a band of overtone of libration vibration (frequency more than that of fundamental intermolecular vibrations) in the water dimer spectrum. This conclusion is in accord with the results of the experiments conducted by the matrix isolation technique [61-64]. The authors of refs. 61-63 have actually noticed a band (3776 c m - 1 ) lying above all the bands of valent vibrations, which in refs. 62 and 63 has been interpreted as a vibration-rotation band. In the 600-500 cm-1 range a solitary band lying far from the bands of other intermolecular vibrations has been noticed by the authors of refs. 63 and 64 who have interpreted it as that of fundamental intermolecular vibrations. Possibly, our results shall cause these interpretations to change. The results of this calculation reveal that for complexes having weak hydrogen bonds the proposed way of allowing for dynamic and electro-optical anharmonicity of molecules enables one to correctly convey the salient features of their spectra which cannot be described by the harmonic model. CONCLUSIONS The possibility of describing the spectra of complexes, having weak and strong hydrogen bonds, within the framework of the unified method of estimating the parameters of dynamic and electro-optical anharmonicity of bonds permits us to think that the established dependences correctly reflect in outline the nature and properties of a hydrogen bond.
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