Hydrogen bonded pyridine N-oxide/trichloroacetic acid complex in polar media: 2D potential energy surface and O–H⋯O vibration analysis using exact vibrational Hamiltonian

Journal of Molecular Structure xxx (2014) xxx–xxx

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Journal of Molecular Structure

journal homepage: www.elsevier .com/ locate /molst ruc

Hydrogen bonded pyridine N-oxide/trichloroacetic acid complex in polarmedia: 2D potential energy surface and O–H� � �O vibration analysis usingexact vibrational Hamiltonian

http://dx.doi.org/10.1016/j.molstruc.2014.02.0160022-2860/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +370 5 2366001; fax: +370 5 2366003.E-mail address: [email protected] (V. Balevicius).

Please cite this article in press as: G. Pitsevich, V. Balevicius, J. Mol. Struct. (2014), http://dx.doi.org/10.1016/j.molstruc.2014.02.016

George Pitsevich a, Vytautas Balevicius b,⇑a Department of Physical Optics, Belarusian State University, Independence av. 4, 220050 Minsk, Belarusb Department of General Physics and Spectroscopy, Vilnius University, Sauletekio 9-3, LT-10222 Vilnius, Lithuania

h i g h l i g h t s

� Exact form of kinetic energy operatorfor 1D and 2D vibrationalHamiltonian was derived.� Numerical solution of 2D Schrödinger

equation using Fourier series wasrealized.� 1D and 2D PES for PyO/TCA complex

in acetonitrile were obtained.� The wave numbers were calculated

and compared with experiment.� Comparison of results obtained using

various sets of vibrational coordinateswas made.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 20 January 2014Received in revised form 7 February 2014Accepted 8 February 2014Available online xxxx

Keywords:Potential energy surface (PES)H-bondInfrared spectroscopyVibrational HamiltonianExact form of kinetic energy operator

a b s t r a c t

The analysis of O–H� � �O stretching vibrations of the pyridine N-oxide/trichloroacetic acid (PyO/TCA)H-bond complex in acetonitrile solution was carried out. 1D and 2D potential energy surfaces associatedwith the variation of valence coordinates of hydroxyl and hydrogen bonds were calculated for this pur-pose in the B3LYP/cc-pVTZ approximation. The exact form of kinetic energy operator was obtained usingthese coordinates and Wilson’s vectors. The numerical solution of 2D Schrödinger equation using Fourierseries was realized and the wave numbers of O–H and O� � �H vibrations were calculated and comparedwith the results obtained using different sets of vibrational coordinates. The values of the O–H� � �Ostretching frequencies obtained as a result of the matrix diagonalization were discussed and comparedwith the experimental data, the results of harmonic- and anharmonic computations as well as withthe results of 3D computations of potential energy surfaces of the PyO/TCA complex using simplified formof kinetic energy operator.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The complexes of carboxylic acids with pyridine N-oxide (PyO)are the promising benchmark systems for the studies of short- and

very short O–H� � �O hydrogen bonding [1–3]. These systems exhibitasymmetric and flat single well potential energy surfaces (PES),which result in large-amplitude proton motion with frequent pro-ton transfer. The interest in H-bonding and proton dynamics is sig-nificantly gained by its importance in modern material science,understanding fine features of organic and enzymatic reactions,transition state, vibrationally enhanced catalysis, etc. [4–7].

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2 G. Pitsevich, V. Balevicius / Journal of Molecular Structure xxx (2014) xxx–xxx

The PyO/TCA (trichloroacetic acid) is a peculiar complex in thisseries having an extremely short donor–acceptor separation of2.430 Å [1]. Recently we have performed for the title complex(PyO/TCA) the calculations of 1D and 2D PES and O–H vibrationsfrequencies in vacuo [8], 3D calculations in vacuo [9], 3D and anhar-monic calculations in polar media (acetonitrile) [10]. It was shownthat the accuracy prediction of the O–H stretching frequency in-creases while going from 1D to 3D model. Carrying out these1D–3D calculations [8–10] we have used a slightly simplified Ham-iltonian and the set of Cartesian coordinates of the hydroxyl hydro-gen atom necessary to describe its motion. It is quite obvious thatpredictive ability of the calculations should increase enlarging thedimensionality of the vibrational problem as well as improving theprecision of presentation of kinetic and potential energy operatorsin the Hamiltonian. On the other hand, it seems that the results ofcalculations should not depend on the choice of vibrational coordi-nates. Therefore the purposes of the present work were: (i) toderive the exact expression for kinetic energy operator in the Ham-iltonian, which describes the vibrational motion of the hydroxylhydrogen atom using several natural (internal) coordinates; (ii)to apply it solving 2D Schrödinger equation by using of Fourier ser-ies; (iii) to calculate 1D and 2D potential energy surfaces and thefrequencies of O–H and O� � �H stretching vibrations in PyO/TCAcomplex in acetonitrile solution and to compare them with theavailable experimental data.

2. The exact form of kinetic energy operator in the Schrödingerequation

Let us to restrict the dimension of the current problem to 1Dand 2D cases and to use the lengths of O–H and O� � �H bonds asthe vibrational coordinates (denoted q and Q, respectively). Consid-ering a restricted dimensionality of the current problem it is clearthat there are three atoms in the bridge O–H� � �O that participate inthe vibrational motion, while in the previously used models [8–10]the positions of all atoms in the H-bond complex were fixed, ex-cept the bridge hydrogen. The Schrödinger equation in the Born–Oppenheimer approximation can be written as

��h2

2MA

@2

@2xA

þ @2

@2yA

þ @2

@2zA

" #W�

�h2

2MB

@2

@2xB

þ @2

@2yB

þ @2

@2zB

" #W

��h2

2MC

@2

@2xC

þ @2

@2yC

þ @2

@2zC

" #Wþ UðxA . . .ÞW

¼ EW; ð1Þ

where A, B, C are understood as O, H, O, respectively, xi, yi and zi arethe Cartesian coordinates of i-th atom, MA, MB, MC are the masses ofcorresponding atoms. We are interested in two internal coordinates,namely: q ¼ lAB � l0

AB and Q ¼ LBC � L0BC , where lAB and LBC are the

lengths of the corresponding bonds and l0AB; L0

BC are the values ofthese lengths in the equilibrium configuration.

The transitions from the Cartesian to internal coordinates in theSchrödinger equation are usually based on the Podolsky transform[11]. This was realized in some recent works [12–15] as well as inthe program package [16]. In the present work trying to get the ex-act expression of the kinetic energy operator in the q(Q) space weuse the chain rule approach [17–19]. It is expected the operator insuch a form to be more convenient for the computing and can beapplied without any additional simplifying assumptions. The deri-vation consists of the following steps.

Since xA affects only q we obtain:

@2

@2xA

¼ @

@xA

@q@xA

@

@q¼ @

2q@x2

A

@

@qþ @q

@xA

� �2@2

@q2 : ð2Þ

Please cite this article in press as: G. Pitsevich, V. Balevicius, J. Mol. Struct. (20

The similar expressions can be constructed for yA and zA, respec-tively. However, xB affects on both – q and Q. Therefore:

@2

@2xB

¼ @2q@x2

B

@

@qþ @

2Q@x2

B

@

@Qþ @q

@xB

� �2@2

@q2 þ@Q@xB

� �2@2

@Q 2

þ 2@q@xB

@Q@xB

@2

@q@Q: ð3Þ

The analogous formulas can be obtained for yB and zB. Hence,

��h2

2MB

@2

@2xB

þ @2

@2yB

þ @2

@2zB

" #W

¼ ��h2

2MB

@2Q@x2

B

þ @2Q@y2

B

þ @2Q@z2

B

" #@

@QW�

�h2

2MB

@Q@xB

� �2

þ @Q@yB

� �2"

þ @Q@zB

� �2#@2

@Q 2 W��h2

2MB

@2q@x2

B

þ @2q@y2

B

þ @2q@z2

B

" #@

@qW

��h2

2MB

@q@xB

� �2

þ @q@yB

� �2

þ @q@zB

� �2" #

@2

@q2 W

��h2

2MB2

@q@xB

@Q@xBþ @q@yB

@Q@yBþ @q@zB

@Q@zB

� �@2

@q@QW;

In the next steps of this treatment we get:

��h2

2MA

@2

@2xA

þ @2

@2yA

þ @2

@2zA

" #W�

�h2

2MB

@2

@2xB

þ @2

@2yB

þ @2

@2zB

" #W

��h2

2MC

@2

@2xC

þ @2

@2yC

þ @2

@2zC

" #W

¼ ��h2

2MA

@2q@x2

A

þ @2q@y2

A

þ @2q@z2

A

" #�

�h2

2MB

@2q@x2

B

þ @2q@y2

B

þ @2q@z2

B

" #" #@

@qW

þ ��h2

2MB

@2Q@x2

B

þ @2Q@y2

B

þ @2Q@z2

B

" #�

�h2

2MC

@2Q@x2

C

þ @2Q@y2

C

þ @2Q@z2

C

" #" #@

@QW

þ ��h2

2MA

@q@xA

� �2

þ @q@yA

� �2

þ @q@zA

� �2" #"

��h2

2MB

@q@xB

� �2

þ @q@yB

� �2

þ @q@zB

� �2" ##

@2

@q2 W

þ � �h2

2MB

@Q@xB

� �2

þ @Q@yB

� �2

þ @Q@zB

� �2" #"

��h2

2MC

@Q@xC

� �2

þ @Q@yC

� �2

þ @Q@zC

� �2" ##

@2

@Q 2 W

þ��h2

MB

@q@xB

@Q@xBþ @q@yB

@Q@yBþ @q@zB

@Q@zB

� �@2

@q@QW: ð4Þ

Using formalism of Wilson’s~s vectors [20] one can obtain:

gradrAðqÞ ¼ @q

@xAþ @q@yAþ @q@zA¼~sq

rA;

@q@xA

� �2

þ @q@yA

� �2

þ @q@zA

� �2

¼ ð~sqrAÞ2; ð5Þ

@2q@x2

B

þ @2q@y2

B

þ @2q@z2

B

¼ div~sqrA: ð6Þ

and substituting (5) and (6) to (4) we get

14), http://dx.doi.org/10.1016/j.molstruc.2014.02.016


G. Pitsevich, V. Balevicius / Journal of Molecular Structure xxx (2014) xxx–xxx 3

��h2

2MAdiv~sq

rA�

�h2

2MBdiv~sq

rB

" #@

@qW

þ ��h2

2MBdiv~sQ

rB�

�h2

2MCdiv~sQ

rC

" #@

@QW

þ ��h2

2MAð~sq

rAÞ2 �

�h2

2MBð~sq

rBÞ2

" #@2

@q2 W

þ ��h2

2MBð~sQ

rBÞ2 �

�h2

2MCð~sQ

rCÞ2

" #@2

@Q 2 W��h2

MB½ð~sq

rB~sQ

rBÞ�@2

@q@QW: ð7Þ

If we take into account that

~s qrA¼~eAB; ~s q

rB¼ �~eAB; ~s Q

rC¼~eBC ; ~s Q

rB¼ �~eBC ; div rA

~eAB ¼2

lAB;

div rC~eBC ¼ �

2LBC

;

we obtain from (7) the following expression:

��h2

2MA

2lAB�

�h2

2MB

2lAB

" #@

@qWþ �

�h2

2MB

2LBC�

�h2

2MC

2LBC

" #@

@QW

þ ��h2

2MA�

�h2

2MB

" #@2

@q2 Wþ ��h2

2MB�

�h2

2MC

" #@2

@Q 2 W

��h2 cos h

MB

@2

@q@QW:

Due to lAB ¼ l0AB þ q; LBC ¼ L0

BC þ Q and simplifying the aboveexpression we get

��h2

2lAB

2

l0AB þ q

" #@

@qWþ �

�h2

2lBC

2L0

BC þ Q

" #@

@QWþ �

�h2

2lAB

" #@2

@q2 W

þ ��h2

2lBC

" #@2

@Q2 W��h2 cos h

MB

@2

@q@QW

¼ ��h2

2lAB

@2

@q2 W��h2

2lBC

@2

@Q 2 W��h2

2lAB

2

l0AB þ q

@

@qW

��h2

2lBC

2L0

BC þ Q

@

@QW�

�h2 cos hMB

@2

@q@QW; ð8Þ

where h is the angle between~eAB and~eBC in the equilibrium config-uration, lAB is the reduced mass of A and B atoms.

The Schrödinger Eq. (1) now takes the form:

��h2

2lAB

@2

@q2 W��h2

2lBC

@2

@Q2 W��h2

2lAB

2

l0AB þ q

@

@qW�

�h2

2lBC

� 2L0

BC þ Q

@

@QW�

�h2 cos hMB

@2

@q@QWþ Uðq;QÞW ¼ EW; ð9Þ

which, introducing the dimensionless coordinates h ¼ q=l0;

H ¼ Q=l0; h0 ¼ l0AB=l0; H0 ¼ L0

BC=l0; where l0 = 1 Å, can be written as:

��h2

2lABl20

@2

@h2 W��h2

2lBCl20

@2

@H2 W��h2

2lABl20

2ðh0 þ hÞ

@

@hW

��h2

2lBCl20

2ðH0 þ HÞ

@

@HW�

�h2 cos h

MBl20

@2

@h@HWþ Uðh;HÞW ¼ EW:

ð10Þ

In the final step denoting FOH ¼ �h2

2lOHl20and FH ¼ �h2

2MBl20we rewrite

(10) as:

� FOH@2

@h2 W� FOH@2

@H2 W� FOH2

ðh0 þ hÞ@

@hW� FOH

2ðH0 þ HÞ

@

@HW

� 2FH cos h@2

@h@HWþ Uðh;HÞW ¼ EW: ð11Þ


This is the target 2D Schrödinger equation with the exact vibra-tional Hamiltonian that contains not only second but also firstderivatives of the wave function respect to the vibrationalcoordinates.

The 1D equation with the exact form of kinetic energy operatoris much easier to obtain:

��h2

2ll20

@2

@h2 þ2

l0AB þ h

@

@h

" #Wþ UðhÞW ¼ EW: ð12Þ

3. Solving of 2D Schrödinger equation by using of Fourier series

Let us to rewrite (11) as follows:

�FOH@2

@h2 W� FOH@2

@H2 W� FOHTðhÞ @@h

W� FOHTðHÞ @@H

W

� 2FH cos h@2

@h@HWþ Uðh;HÞW

¼ EW: ð13Þ

We solve the Eq. (13) applying the method that was used earlier[21,22]. The functions therein can be presented as follows:

Wðh;HÞ ¼Xk;l¼0

Wk;leiðxkhþmlHÞ; ð14Þ

Uðh;HÞ ¼X

s;t

Us;teiðxshþmtHÞ; ð15Þ

TðhÞ ¼X

m

Tmeixmh; ð16Þ

TðHÞ ¼X

n

TneimnH; ð17Þ

where x ¼ 2pl0d ; m ¼ 2pl0

D ; the quantities d and D are the linear dimen-sions of the rectangle of the change of the coordinates q and Q,respectively. The calculations of the potential energy are carriedout on the nodes of a certain grid in this area.

By substituting (14)–(17) to (13) we obtain:

Xk;l¼0

FOHx2k2Wk;leiðxkhþmlHÞ þXk;l¼0

FOHm2l2Wk;leiðxkhþmlHÞ

� FOH

Xm

TmeixmhXk;l¼0

ixkWk;leiðxkhþmlHÞ

� FOH

Xn

TneimnHXk;l¼0

imlWk;leiðxkhþmlHÞ

þ 2FH cos hXk;l¼0

xkmlWk;leiðxkhþmlHÞ

þX

s;t

Us;teiðxshþmtHÞXk;l¼0

Wk;leiðxkhþmlHÞ ¼ EXk;l¼0

Wk;leiðxkhþmlHÞ: ð18Þ

This expression can be transformed as follows:Xk;l¼0

ðFOHðx2k2 þ m2l2Þ þ 2 cos hFHxmklÞWk;leiðxkhþmlHÞ

�X

k;l;m¼0

iFOHxkTmWk;leiðxðkþmÞhþmlHÞ

�X

k;l;n¼0

iFOHmlTnWk;leiðxkhþmðlþnÞHÞ

þX

k;l;s;t¼0

Us;tWk;leiðxðkþsÞhþmðlþtÞHÞ

¼ 0: ð19Þ

The grouping of the coefficients of the exponents for the modesk0 and l0 leads to:




Xk;l¼0

ðFOHðx2k02 þ m2l02Þ þ 2 cos hFHxmk0l0ÞWk;leiðxk0hþml0HÞ

�X

k;l;m¼0

iFOHxkTk0�kWk;l0eiðxk0hþml0HÞ

�X

k;l;n¼0

iFOHmlTl0�lWk0 ;leiðxk0hþml0HÞ

þX

k;l;s;t¼0

Uk0�k;l0�lWk;leiðxk0kþml0HÞ

¼ 0: ð20Þ

The diagonal elements of the Hamiltonian matrix can be thenexpressed as

Hðk0 l0Þðk0 l0 Þ ¼ FOHðx2k02 þ m2l02Þ þ 2 cos hFHxmk0l0 þ U00

� iFOHT0ðxk0 þ ml0Þ; ð21Þ

and the non-diagonal ones as

Hðk0 l0Þðkl0Þ ¼ Uk0�k;0 � iFOHxkTk0�k; ð22Þ

Hðk0 l0Þðk0 lÞ ¼ U0;l0�l � iFOHmlTl0�l; ð23Þ

Hðk0 l0ÞðklÞ ¼ Uk0�k;l0�l; ð24Þ

respectively. The Hamiltonian matrix is composed using the ob-tained expressions (21)–(24) and the stationary energy levels andthe wave functions of the corresponding vibrational states aredetermined by its diagonalization.

4. Calculations of 1D and 2D potential energy surfaces

For comparison of the results of the current study with thosecalculated earlier, 1D and 2D potential energy surface (PES) calcu-lations were carried out in the approximation B3LYP/cc-pVTZ. Thislevel of theory was used also in [8–10]. The computations wereperformed using Gaussian 09 quantum-chemistry package [23].The structure of the complex PyO/TCA in acetonitrile is shown inFig. 1.

The energy values were calculated with the spacing of 0.1 Å be-tween the nodes on the rectangular grid of the linear dimensionsd � D = 2.1 � 3.1 Å2. The variation boundaries of the coordinateswere used Dh e [�0.5; 1.6] and DH e [�1.; 2.1]. The analytical rep-resentation of potential energies U(h); U(H) and U(h, H) were madeby the third order spline interpolation using Mathematica package[24]. The corresponding PES are shown in Figs. 2 and 3.

Fig. 1. The structure of PyO/TCA complex with indication of atomic numbers and Carteconformers see Ref. [10].


The expansion coefficients Us,t were determined using the ob-tained analytical representation (15) and Mathematica package.Since T(h) and T(H) were also presented analytically (16), (17), itwas easy to determine Tm and Tn. Then the matrix Hi,j elementswere computed varying matrix dimensions from 900 � 900 to1600 � 1600. The stabilization of the values of some of the mostdeeply-lying energy levels was achieved increasing the matrixdimensions. This allowed to determine the frequencies of the fun-damental vibrations of O–H and O� � �H bonds. The results of the cal-culations are presented in Table 1.

The values of the O–H stretching frequency in the acetonitrilesolution were found 1070, 1041, 1005 and 350 cm–1 using 1D,2D, 3D and anharmonic calculations, respectively, whereas theexperimental value is close to �950 cm–1 [8–10]. It is obvious thatanharmonic approach unable to predict correctly the O–H stretch-ing frequency in this case.

5. Discussion of the results

At first sight the data presented in Table 1 seems to be contra-dict to the results of the earlier calculations using the simplifiedHamiltonian [10]. But they are not. It is obvious that the calcula-tions using 1D model and the internal coordinates h and H cannotsatisfactory predict values of the frequencies of correspondingvibrations. According to Fig. 2, the potential curves have a shapetypical for ordinary chemical bond (one of them (H) is somewhatweaker than the other (h)). This is mainly due to the increase ofthe strength of H-bond is always attended by the decrease of thecovalent O–H bond. It is also clear that the major role determiningthe frequencies of the stretching vibrations for both bonds in theH-bond complex plays their kinematic interaction. It is completelyignored in 1D model and taken into account when 2D model isused. And indeed, as it follows from Table 1, the application of2D models significantly improve the results of the calculations.At the same time it is worth to note that the interaction betweentwo oscillators is reproduced in 1D model better if instead of theinternal coordinates the Cartesian ones are used (Table 1 and Ref.[10]) for the description of the bridge hydrogen movement. Letus consider this situation in more details.

It could be thought that the treatment of 1D motion of thebridge proton (H13, Fig. 1) along X, i.e. the axis directed alongO–H� � �O bond, and that should lead to the approximately equalresults determining the frequency of O–H stretching using thecoordinates h and H in 2D approach. In fact the difference is signif-icant (1070 and 1224 cm�1, Table 1), but it is not as essential as

sian axes. For more details of the optimized hydrogen bonding geometry and the



Fig. 2. Potential energy of the PyO/TCA complex in acetonitrile as a function of the dimensionless internal coordinates h (blue) and H (red). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. 2D PES of the PyO/TCA complex in acetonitrile as a function of the dimensionless internal coordinates h and H and its intersection with the plane corresponding to theH-bond proton (H13, Fig. 1) motion along X axis.

Table 1The values of calculated and experimental frequencies (all in cm–1) of O–H and O� � �H stretching vibrations in PyO/TCA complex in acetonitrile solution using 1D and 2D modelswith Cartesian (X, Y, Z) and internal (h, H) coordinates.

Vibrational mode Approximation used for the frequency calculation

1DX 1Dh 1DH 2DXY 2DXZ 2DhH Anharmonic* Experimental value*

O–H 1070 2569 – 1054 1036 1224 350 �950O� � �H – – 1094 – – 498 132 –

* See Refs. [8,10].


comparing the results of 1D calculations using the Cartesian andinternal coordinates (1070 and 2569 cm�1, ibid). The reason forthis can be understood perceiving that in the first case motion ofH13 practically corresponds to the normal mode of hydroxyl groupvalence vibration. In this case the set of coordinates (h, H) becomesredundant. Indeed, as one can see from Fig. 1,

H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH0

OOÞ2 þ ðh0

OH þ hÞ2� 2 cos bH0

OOðh0OH þ hÞ

q� H0

OO; ð25Þ

where H0OO ¼ 2:48148 is the equilibrium dimensionless distance be-

tween O14 and O11, b! \H13O14O11 ¼ 5:02239 deg, h0OH ¼ 1:05912


is the equilibrium dimensionless distance between O14 and H13.Due to the fact that b is close to zero, H (as function of h) is almoststraight line in the (h, H) plane. The curve obtained as the result ofthe intersection of the plane, which contains this line and is perpen-dicular to the (h, H) plane, with the U(h, H) surface (see Fig. 3), is theclosest one to the potential curve U(X) determined in [10]. Howeverthe motion along this curve on 2D surface does not fully correspondin terms of energy to the motion of H13 along the coordinate X. In-deed, in the last case, besides the coupled changing of the lengths ofO–H and O� � �H bonds, the angle h between them (i.e. the in planebending coordinate) is also changing. In the first case the value of



Fig. 4. Difference DU between U(h) and U(X) potential energy curves along the coordinate X, i.e. along O–H� � �O bridge.


this angle is staying invariable that cannot represent the actualdynamics of vibrations of the hydroxyl group. It is obvious thatexactly the last circumstance is crucial determining the reason ofthe difference between calculated frequencies when using two ap-proaches and, in particular, smaller accuracy in comparison withexperimental value using h and H coordinates in 2D model. We havecalculated the energy along the U(h) path. The difference DU =U(h) � U(X) along the coordinate X is represented in Fig. 4.

The wave number of mOH have been calculated using U(h) poten-tial curve and the simplified Hamiltonian from [10]. In this case thefrequency of mOH vibration really increases to 1171 cm–1. It shouldmean that the way of representing of the potential energy is moreessential than the corrections to the energy due to more accuratedefinition of kinetic energy operator. Increasing the dimensionalityof the vibrational problem to 4D using the natural coordinates andby taking into account the in plane- and out of plane O–H� � �Obending coordinates will definitely lead to the further improve-ment of the precision of mOH calculations.

6. Conclusions and outlooks

The exact form of kinetic energy operator for 1D and 2D vibra-tional Hamiltonian in internal coordinates was obtained using Wil-son’s~s vectors.

The numerical solution of 2D Schrödinger equation using Fou-rier series was realized and 1D and 2D PES for PyO/TCA complexin acetonitrile solution were obtained. The wave numbers of O–Hand O� � �H vibrations were calculated and compared with theexperimental data and the results obtained using different sets ofvibrational coordinates.

The PES calculations at the points of the configuration space sig-nificantly distant from the equilibrium point is an urgent task thatpromotes improvement of calculation results of the frequencies oflarge amplitude vibrations, irrespective of the choice of coordi-nates. At the same time, successful selection of the last ones givesthe opportunity to minimize the efforts associated with computercalculations of PES while maintaining good predictive ability ofcomputational results. It is very important to be able to comparethe results of calculations performed using different sets of vibra-tional coordinates. The increase of the possibilities of computersallows to increase the dimensionality of the problem and demandsmore accurate presentation of kinetic energy operator whiledescribing vibrations of molecular systems. In this regard, the


exact form of vibrational Hamiltonian in the convenient form de-rived in the present work is also of great interest for furthernumerical solution of Schrödinger equation solving specific prob-lems in other H-bonded systems.

Acknowledgements

The State Foundation for Basic Research of Belarusian Republic(Grant No F13K-064) and the funding from the European Commu-nity’s social foundation under Grant Agreement No VP1-3.1-ŠMM-08-K-01-004/KS-120000-1756 are acknowledged. Prof. Dr. JanezMavri and Prof. Dr. Zdzislaw Latajka are thanked for commentsand stimulating discussions. The calculations were performedusing the computational facilities of the joint computational clus-ter of SSI Institute for Single Crystals and Institute for ScintillationMaterials of National Academy of Science of Ukraine incorporatedinto Ukrainian National Grid.

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