Complex Symmetrized Calculations on Ammonia Vibrational Levels

CEJC 3(3) 2005 556–569

Complex symmetrized calculations on ammonia vibrational levels

Svetoslav Rashev∗, Lyubo Tsonev, Dimo Z. Zhechev Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussee 72, 1784 Soﬁa, Bulgaria

Received 1 February 2005; accepted 20 May 2005

Abstract: This paper introduces a fully symmetrized Hamiltonian formalism designed for description of vibrational motion in ammonia (and any XH3 molecule). A major feature of our approach is the introduction of complex basis vibrational wavefunctions in product form, satisfying the complex symmetry species (CSS) of the molecular symmetric top point group (D3h). The described formalism for ammonia is an adaptation of the approach, previously developed and applied to benzene, based on the CSS of the point group D6h. The molecular potential energy surface (PES) is presented in the form of a Taylor series expansion around the planar equilibrium state. Using the described formalism, calculations have been carried 14 out on the vibrational overtone and combination levels in NH3 up to vibrational excitation energies corresponding to the fourth N-H stretch overtone. The results from the calculations are adjusted to experimentally measured data, in order to determine the values of the harmonic and some anharmonic force constants of the molecular PES. c Central European Science Journals. All rights reserved.

Keywords: symmetric top point groups, complex symmetry species, ammonia, vibrational overtones

1 Introduction

Ammonia is an important molecule because of its abundance both on earth as well as in the atmospheres of the outer planets, Jupiter and Saturn [1]. Knowing its detailed ro-vibrational energy level structure, the planetary atmospheric temperatures could be determined [1]. For this reason, extensive spectroscopic measurements have been per- 14 formed on NH3 as well as on its isotopomers [1-6], in a wide spectral range. Many

∗ E-mail: [email protected]; Fax: 00 359 2 975 36 32 S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 557 theoretical studies have also been carried out to rationalize the extremely complicated

NH3 PES and ro-vibrational spectrum [2,3,7-26]. Ammonia is a symmetric top molecule, belonging to the point group D3h [1,7,27]. In addition to the floppiness of the molecule is associated with the group of three high frequency and strongly anharmonic N-H stretching modes. This major difficulty for the theoretical treatment requires local mode (LM) treatment, instead of the conventional normal mode (NM) analysis. The N-H stretch overtone structure at the higher excitation energies is additionally complicated through the occurence of a number of strong Fermi and Darling-Dennison resonances [2,3,21,22]. Within this range, the density of the molecular vibrational level increases to the extent that the vibrational computations become difficult and time consuming. In a recent work series [28-31], we introduced the concept of complex symmetry species (CSS) to describe the transformation properties of symmetric top point groups (and particularly the benzene group, D6h), instead of the conventional real symmetry species [1,7,27]. This allowed to greatly reduce the computational effort for large scale vibrational calculations in symmetric top molecules, and particularly for benzene [31]. This paper focuses on adapting and applying the complex symmetrized approach to the ammonia molecule based on the definition of CSS for the ammonia symmetric top point group

D3h. The (LM) N-H stretch part of the basis wavefunctions needs a special symmetrization scheme that has been developed with speciﬁc detail. The present work expands the molecular PES in a Taylor series around the planar equilibrium conﬁguration. The interaction of Hamiltonian terms which couple the separate vibrational degrees of feedom and which incorporate the terms from the second to sixth order, have been taken into account.

2 Complex symmetry species and complex symmetrized curvilinear vibrational coordinates for ammonia

In its ground electronic state, ammonia has two potential minima (where it is pyramidal and of C3v symmetry) that are located symmetrically with respect to the planar state. In the present work we shall use the planar conﬁguration (of D3h symmetry) as a reference point for the series expansion of the molecular PES in terms of the vibrational coordinates. We shall use complex (polar) coordinates to describe the vibrational motion in ammonia [27,35], that demonstrates speciﬁc symmetry properties. Please see refs. 30, 31 for a general discussion of the concept of complex symmetry species for symmetric top point groups. The basic asset of CSS is that they allow the straightforward construction of quantum mechanical basis sets in product form, which are particularly suitable to describe those molecules belonging to a symmetric top point group. Such a set is most helpful to explore highly excited molecular vibrational levels, in the range of excessively high vibrational level density, as demonstrated in our recent work on benzene [28-31]. The introduction of complex symmetry species for the symmetric top point group of ammonia

D3h and their transformation properties and multiplication rules, are displayed in Table 1. As a comparison, the conventional characters of the irreducible representation (symmetry 558 S.Rashevetal. /CentralEuropeanJournalofChemistry3(3) 2005 556–569 species) of this group, are reproduced in Table 2.

Table 1 can be understood as follows: each 2-D symmetry species E of D3h can be effectively considered as two separate 1-D species - Ea and Eb, represented by a pair of c.c basis functions. A new CSS A (A’ or A”) is introduced, that can be either A1, or A2, or A1+iA2, where A1 and A2 are ortho-normal basis functions. In Table 1, for each of the basic symmetry operations (transformations) from D3h, a phase factor is given, in addition to performing complex conjugation (ψ ψ*), to be applied in some of the cases. If the → quantum mechanical basis functions are appropriately defined in complex symmetrized form, as shown below, the multiplication of any two such functions invariably yields a new function, that again has a well defined CSS. Relevant multiplication rules for the CSS of

D3h are summarized at the end of Table 1. These are compared to the conventional rules for multiplication of real 2-D symmetry species, displayed at the end of Table 2. According to the latter ones, e.g. E E = E + A + A , the multiplication of two wavefunctions × 1 2 belonging to the E species, does not yield a wavefunction of a well deﬁned symmetry species, but in general a wavefunction, belonging to a reducible representation of the symmetry group. This feature, inherent to the 2-D real symmetry species, precludes the symmetry product reproducibility, characteristic of the abelian point groups, and makes the symmetrization of quantum mechanical basis sets a rather diﬃcult task. The description of vibrational motion in ammonia, is usually based on the stretchings r of three bond lengths R and the distortions θ of three interbond angles α (r = R R , i i i i i i− 0 θ =α α ), i=1,2,3, where R - equilibrium bond length and α - equilibrium angle i i− 0 0 0 [1,7,27]. The three complex symmetrized N-H stretching coordinates q1, q3a, q3b (modes

#1 and #3) and H-N-H angle distortion coordinates q2, q4a, q4b (modes #2 and #4), can be deﬁned in the following complex symmetrized form [the relevant CSS of D3h (Table 1) are indicated]:

′ ′ q1(A = A1)=(r1 + r2 + r3)/√3, ′ −2 2 q3a(Ea)=(r1 + ε r2 + ε r3)/√3, ′ 2 −2 q3b(Eb)=(r1 + ε r2 + ε r3)/√3, ′′ ′′ q2(A = A1) = R0(θ1 + θ2 + θ3)/√3, (1) ′ −2 2 q4a(Ea) = R0(θ1 + ε θ2 + ε θ3)/√3, ′ 2 −2 q4b(Eb) = R0(θ1 + ε θ2 + ε θ3)√3,

iπ/3 where ε = e . The deﬁnition of q2 is supplemented by the additional condition that q2 should have a negative value when the N-atom is below the H3 plane and positive above.

An arbitrary product of qi coordinates necessarily transforms as one of the CSS of

D3h and can easily be determined from the product rules in Table 1. This latter quality, which is not possessed by the conventional real symmetrized (curvilinear) coordinates Si [1,7,27], is indispensable for the construction of a symmetrized set of basis Hamiltonian eigenfunctions in product form for ammonia. S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 559

3 Zeroth-order Hamiltonian and symmetrized basis set in product form.

The zeroth order Hamiltonian H0 for the present treatment (in fully symmetrized form), takes the form: NH (4) (2) H0 = H0 + H0 (q4a, q4b)+ H0 (q2). (2) NH H0 is the N-H stretch Hamiltonian, a collection of three identical Morse oscillators:

3 2 2 2 NH h¯ ∂ −asri H0 = grr 2 + Dr 1 e . (3) "− 2 ∂ri − # Xi=1 where grr=1/mN +1/mH (mN ,mH - mass of N and H atoms respectively), ar - anharmonic parameter, Dr- dissociation energy. The energy levels of a Morse oscillator are given by:

E = ω (n +1/2) x (n +1/2)2, (4) n NH − NH where ωNH = √frrgrr/(2πc) - harmonic frequency, frr – harmonic force constant, xNH 2 =ωNH(arKr) /2- anharmonic constant, and Kr = h¯ grr/frr. The zeroth-order Hamiltonian for the #4 moder (E’),q is given in the conventional form for a 2-D quantum mechanical harmonic oscillator in terms of polar coordinates [27,36]:

H(4)(q , q )= ω (q q /K2 K2∂2/∂q ∂q ), (5) 0 4a 4b 4 4a 4b 4 − 4 4a 4b −1 ω4 = F4,4G4,4/(2πc) [cm ] is the harmonic frequency of mode #4, K4 = h¯ G4,4F4,4, r F4,4- diagonalq harmonic force constant and G4,4- diagonal symmetrized Wilson’sqG-matrix element [27]. The eigenvalues En4, of the 2-D harmonic oscillator eigenfunctions (n4a,n4b), designated by their quantum numbers n4a, n4b = 0, 1, 2,.. [27,36], are En4 = ω4 (n4 + 1),

(n4=0, 1, ..), n4 = n4a + n4b. All (n4a,n4b) eigenfunctions arise in complex symmetrized form. E.g., (1,0) and (0,1) are of CSS Ea’ and Eb’ respectively, (1,1) is A = A1’, (2,0) and (0,2) are a c.c. pair of CSS Eb’ and Ea’ respectively, (3,0) and (0,3) are a c.c. pair of CSS A’=A iA , etc. 1± 2 To zeroth order, the #2 (inversion) mode is described by a harmonic oscillator Hamil- tonian: H(2)(q )= ω ( K2∂2/∂q2 + q2/K2)/2, (6) 0 2 2 − 2 2 2 2 −1 where ω2 = F2,2G2,2/(2πc) [cm ] is the frequency of the symmetrized mode #2, K2 = q h¯ G2,2F2,2, F2,2- diagonal harmonic force constant and G2,2- diagonal symmetrized r q (2) Wilson’s G-matrix element [27]). The eigenfunctions of H0 will be denoted as n2 . NH | i The nonsymmetrized eigenfunctions of H0 are obtained as products of 3 Morse oscillator eigenfunctions pkl = p k l , corresponding to p quanta excitation in oscillator | i 1 2 3 #1, k quanta in #2 and l quanta in #3; the zeroth-order energy of such a configuration is obtained as the sum of three terms: Ep + Ek + El. NH The quantum mechanical treatment is greatly reduced if the eigenfunctions of H0 are obtained in symmetrized form. The symmetrization process is extremely cumbersome 560 S.Rashevetal. /CentralEuropeanJournalofChemistry3(3) 2005 556–569 when real symmetrized vibrational coordinates are used [20-23]. To derive complex symmetry adapted orthogonal N-H stretch wavefunctions ϕ corresponding to a configuration p1k2l3, simple linear combinations must be taken, by rotating the original configura- L −1/2 tion: ϕ = L Ckpakblc. We introduce the following notation for such a state: ϕ k=1 = L;S;p k l ). Here,P L – normalization factor (possible values are: L=1,3,6), S – the rel- | 1 2 3 evant CSS of D3h. Several symmetrization patterns are possible for ϕ, and can easily be incorporated into a simple algorithm. We shall only give a few examples, illustarting the method of the symmetrization applied. The m=1 (m=p+k+l) overtone contains three states: 3; A′ ;1 ) = (1 +1 +1 )/√3, | 1 1 1 2 3 3; E′ ;1 ) = (1 + ε−21 + ε21 )/√3, | a 1 1 2 3 3; E′ ;1 ) = (1 + ε21 + ε−21 )/√3. | b 1 1 2 3 In the m=2 overtone there are: ′ 3; A1;21) = (21 +22 +23)/√3, | ′ − 3; E ;2 ) = (2 + ε 22 + ε22 )/√3, | a 1 1) 2 3 3; A′ ;1 1 ) = (1 1 +1 1 +1 1 )/√3, | 1 1 2 2 3 3 1 1 2 3; E′ ;1 1 ) = (1 1 + ε−21 1 + ε21 1 )/√3, | a 1 2 2 3 3 1 1 2 3; E′ ;1 1 ) = (1 1 + ε21 1 + ε−21 1 )/√3, etc. | b 1 2 2 3 3 1 1 2

Ultimately, the basis wavefunctions k (eigenfunctions of H ) are obtained as simple | i 0 products of three factors k = L;S;p k l ) (n ,n ) n , in fully (complex) sym- | i | 1 2 3 × 4a 4b ×| 2i metrized form.

4 Interaction hamiltonian

In addition to the (harmonic oscillator) zeroth-order Hamiltonian H0, it is necessary to introduce a considerable number of interaction Hamiltonian terms Hint (all of them in explicit complex symmetrized form) brieﬂy described below. First, the zeroth order harmonic potential for the #2 mode has to be supplemented by a number of terms, which describe the double minimum shape. For this purpose, we have introduced the following corrections that allow the analytical calculation of the matrix elements: H(2)(q ) = h¯2[q2∂2G /∂q2](∂2/∂q2)+ f (4)q4, 4 2 − 2 2,2 2 2 2 H(2)(q ) = h¯2(q4∂4G /∂q4)(∂2/∂q2)+ f (6)q6, (7) 6 2 − 2 2,2 2 2 2 V (2)(q ) = Bexp( dq2/K2)+ B exp( d q2/K2) b 2 − 2 2 1 − 1 2 2 +B exp( d q2/K2)+ B exp( d q2/K2). 2 − 2 2 2 3 − 3 2 2 The ﬁrst two terms are (small) corrections to the overall shape of the harmonic potential curve, while the third term describes the double minimum behavior, by means of the S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 561

variable parameters B, d, B1, d1, etc. The G-matrix derivatives, displayed in the above equation and in the one below, have been precisely calculated in symmetrized form. The following are terms describing the bilinear, cubic, quartic and some sextic couplings of the three N-H stretch oscillators:

2 2 2 2 Hr,r = h¯ g1,2(∂ /∂r1∂r2 + ∂ /∂r2∂r3 + ∂ /∂r3∂r1)+ f1,2(r1r2 + r2r3 + r3r1), − 2 2 2 Hr,r.r = f1,2,3r1r2r3 + f1,1,2[(r1(r2 + r3)+ r2(r1 + r3)+ r3(r1 + r2)], 2 2 2 2 2 2 2 2 2 Hr,r,r,r = f1,1,2,2(r1r2 + r2r3 + r3r1)+ f1,1,2,3(r1r2r3 + r2r3r1 + r3r1r2) (8) 3 3 3 +f1,1,1,2[r1(r2 + r3)+ r2(r1 + r3)+ r3(r2 + r1)], 2 2 2 3 3 3 3 3 3 Hr,r,r,r,r,r = f1,1,2,2,3,3r1r2r3 + f1,1,1,2,2,2(r1r2 + r2r3 + r3r1), where f1,2 is a quadratic nondiagonal harmonic force constant; g1,2=cos(α0)/mN ; f1,2,3, f1,1,2 are cubic, f1,1,2,2, f1,1,2,3, f1,1,1,2- quartic, f1,1,2,2,3,3, f1,1,1,2,2,2 - sextic nondiagonal force constants. In addition to these terms, Hint incorporates the interaction of the N-H stretch system with the symmetrized 2-D mode #4:

H = h¯2G (∂2/∂q ∂q + ∂2/∂q ∂q )+ F (q q + q q ). (9) 3,4 − 3,4 3a 4b 4a 3b 3,4 3a 4b 3b 4a

F3,4 and G3,4=3√3/(2mN ) are the symmetrized nondiagonal harmonic force constant and G-matrix element respectively, connecting the two modes. We have also included in Hint two cubic interaction terms, incorporating two cubic nondiagonal force constants, F3,4,4 and F1,4,4:

(3) 2 2 2 H3,4,4 = h¯ (q1∂G3,4/∂q1)(∂ /∂q3a∂q4b + ∂ /∂q4a∂q3b) − 2 2 h¯ (q1∂G4,4/∂q1)(∂ /∂q4a∂q4b) (10) − 2 2 +F3,4,4(q3aq4a + q3bq4b)+ F1,4,4q1q4aq4b. Finally, there are cubic and quartic terms, describing the coupling of the inversion mode #2 with the N-H stretch system (modes #1 and #3) and the asymmetric bend, mode #4:

H(3) = h¯2(q ∂G /∂q )(∂2/∂q2)+ F q q2. (11) 1,2,2 − 1 2,2 1 2 1,2,2 1 2 (4) 2 2 2 2 2 2 H3,2,2,4 = h¯ (q2∂ G3,4/∂q2 )[∂ /∂q3a∂q4b + ∂ /∂q4a∂q3b] − 2 2 h¯ [q3aq4b∂ G3,4/∂q3a∂q4b − 2 2 2 +q3bq4a∂ G3,4/∂q3b∂q4a](∂ /∂q3a∂q4b + ∂ /∂q3b∂q4a) (12) 2 +F3,2,2,4q2 (q3aq4b + q3bq4a).

5 6-D computations on the ammonia vibrational system

This work carries out full-scale 6-D computations on the vibrational system of ammonia 14 −1 NH3 at excitation energies up to about 15500 cm . The completely nonperturbative computational approach which was applied, included an artiﬁcial intelligence search algorithm [32,33] for the selection of an eﬀective active space of N relevant basis functions k | i 562 S.Rashevetal. /CentralEuropeanJournalofChemistry3(3) 2005 556–569

(eigenstates of H0). All of them belong to one and the same CSS. Simultaneously, the (real symmetric) Hamiltonian matrix Hi,k of the vibrational problem is being composed, whose diagonal elements are k H + Hint k and the oﬀ-diagonal elements of which are given h | 0 | i by k Hint k′ . For this procedure, as described in detail in our previous work [30,31], the h | | i availability of a symmetrized separable vibrational basis set L;S;p k l ) (n ,n ) n | 1 2 3 × 4a 4b | 2i is of crucial importance. Next, Hi,k is diagonalized making use of a Lanczos tridiagonal- ization routine [34], thus obtaining the molecular vibrational levels. In this calculation, the values of the harmonic and anharmonic force constants, part of the expressions of int both H0 and H , were regarded as variable parameters: frr, F2,2, F4,4, [harmonic diagonal force constants for the local N-H stretch oscillator, the #2 (inversion) mode and the #4 (asymmetric bend)], xNH (anharmonic correction for the N-H stretch, related to the cubic anharmonic constant of the Morse potential), f (4) and f (6) (quartic and sextic corrections to the overall shape of the inversion mode potential), B, B1, B2, B3, d, d1, d2, d3, (parameters, characterizing the double minimum shape of the inversion potential), f1,2, f1,2,3, f1,1,2, f1,1,2,2, f1,1,2,3, f1,1,1,2, f1,1,2,2,3,3, f1,1,1,2,2,2 (harmonic, cubic, quartic and sextic nondiagonal force constants, responsible for the couplings among the three N-H stretch oscillators), F3,4, F1,4,4, F3,4,4, F1,2,2, F3,2,2,4 (symmetrized bilinear, cubic and quartic nondiagonal force constants, describing the couplings between the symmetrized modes #1, #2, #3 and #4). These (harmonic and anharmonic) force constant parameters were varied (using a least squares routine), until the calculated energy levels were obtained as close as possible to the experimentally measured values. The obtained results for the calculated vibrational levels of ammonia are displayed in Table 3. The experimentally measured frequences have also been given in the Table for comparison. Each level is designated by the symmetrized basis state L;S;p k l ) (n ,n ) n , | 1 2 3 × 4a 4b | 2i giving the predominant contribution to relevant the vibrational eigenstate. It can be seen, that a satisfactory agreement has been achieved between the theoretically calculated and the experimentally measured frequencies. This agreement was obtained at following force constant parameter values, that are summarized in Table 4.

6 Conclusion

The main purpose of the paper was to give a concise description of a completely symmetrized quantum mechanical Hamiltonian formalism and algorithm, designed to improve and extend the possibilities of large scale vibrational calculations on ammonia (and in general for XH3 molecules). The most important feature of the described approach is the definition and employment of complex symmetrized (curvilinear) vibrational coordinates and wavefunctions that transform according to the complex symmetry species of the symmetric top point group D3h. The PES for the ground electronic state of ammonia was defined as a Taylor series expansion in the employed vibrational coordinates at the planar equilibrium molecular configuration. Using the described symmetrized Hamiltonian formalism, 6-D calculations have been carried out on the vibrational levels of ammonia 14 NH3. Reasonable agreement between the calculated and measured frequencies could S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 563 be obtained, for levels up to the range of the fourth N-H stretch overtone. This can be achieved by varying the harmonic and anharmonic force constants, as input parameters. Work is in progress, to extend these calculations to higher excited vibrational states, as well as to the other symmetric ammonia isotopomers.

Acknowledgment

Special thanks to the Bulgarian Ministry of Education and Scientiﬁc Research Grant (1415) for funding.

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D c c ’(ψ ψ*) σ σ (ψ ψ*) 3h 3 2 → h v → ′ ′ ′ A = A1 iA2 1 1 1 1 ′′ ′′ ± ′′ A = A1 iA2 1 1 1 1 ′ ± 2 − − Ea εg 1 1 1 ′ 2 Eb εg 1 1 1 E′′ εg2 1 1 1 a − − E′′ εg2 1 1 1 b − − S A = S, E A = E, E E = A + A + E, ‘ ‘ = ‘, ‘ ”=“,“ ” = ‘, × 1 × 1,2 × 1 2 × × × A A = A , A A = A (S-any symmetry). 1× 2 2 2 × 2 1 Table 1 Complex symmetry species, their transformation rules, and their multiplication rules for the symmetric top point group D3h;c3 – rotation around the top axis by 2π/3, c2’ - rotation by π around an axis, perpendicular to the top axis of the molecule, σ v - reflection in a vertical iπ/3 plane, σ h - reflection in the molecular plane, ε = e ; the notation ψ ψ* signifies, that in addition to being multiplied by the indicated phase factor, the function should→ also be complex conjugated. S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 567

D6 I c3 c2 σh σv

A1’ 11 1 1 1 A ’ 1 1 1 1 1 2 − − A ” 1 1 1 1 1 1 − − A ” 1 1 1 1 1 2 − − E’ 2 1 0 1 0 − − E” 21 0 1 0 S A = S, E A = E, E E = A + A + E, ‘ ‘ = ‘, ‘ ”=“,“ ” = ‘, × 1 × 1,2 × 1 2 × × × A A = A , A A = A (S-any symmetry). 1× 2 2 2 × 2 1 Table 2 Character table for the point group D3h (gIg gdentity operation); multiplication of symmetry species. − 568 S.Rashevetal. /CentralEuropeanJournalofChemistry3(3) 2005 556–569

Level n2=1(A2”) n2=2(A1’) n2=3(A2”) n2=4(A1’) n2=5(A1”) (Symmetry) GS(-) ν (+) ν ( ) 2ν (+) 2ν ( ) 2 2 − 2 2 − calculated 0.555 932.35 967.76 1597. 72 1882.79 experimental 0.79 [16] 932.43 [16] 968.12 [16] 1597.470 [4] 1882.179 [4]

Level n2=6(A1’) n2=7(A2”) n2=8(A1’) (Symmetry) 3ν (+) 3ν ( ) 4ν (+) 2 2 − 2 calculated 2383.03 2895.84 3461.78 experimental 2384.15 [5] 2895.52 [5] 3462 [6]

Level n =1) (E ’) n =1, 3;A ’;1 ) 3;A ’;1 ) 3;E ’;1 ) | 4a a | 4a | 1 1 | 1 1 | a 1 (Symmetry) n =1) (E ”) n =1) (A ”) 2 a | 2 2 calculated 1626.5 1627.4 3336.3 3337.0 3443.6 experimental 1626.28 [4] 1627.37 [4] 3336.08 [6] 3337.11 [6] 3443.677 [6]

Level 3;E ’;1 ) 3;A ’;2 ) 3;E ’;2 ) 3;A ’;1 1 ) | a 1 | 1 1 | a 1 | 1 1 2 (Symmetry) n =1) (E ”) | 2 a calculated 3444.2 6606.3 6609.5 6796.4 experimental 3443.99 [6] 6606.0 [3] 6608.83 [3] 6795.3 [3]

Level 3;E ’;1 1 ) 1;A ;1 1 1 ) 3;E ;3 ) 3;A ;5 ) | a 1 1 | 1 1 2 3 | a 1 | 1 1 calculated 6852.4 10237.2 9693.1 15456.4 experimental 6850.2 [3] 10234.73 [3] 9689.84 [3] 15451.19 [3]

Table 3 Calculated fundamental and overtone frequencies (in cm−1) for the N-H stretch vibra- 14 tional system of ammonia NH3, compared with the experimentally measured values. Employed notation L;S;m1n2k3): m1n2k3 denotes the level of excitation for the three N-H oscillators, S (CSS) and| L – normalization factor. The more conventional notation for the inversion levels is also given. S. Rashev et al. / Central European Journal of Chemistry 3(3) 2005556–569 569

(4) force cte. frr xNH F2,2 F4,4 f

description LocalN-H LocalN-H mode#2 mode#4 mode#2 stretch, stretch, harmonic harmonic quartic harmonic anharmonic value 6.965 75.526 0.1032 1.3487 0.07562 − unit aJ/A2 cm−1 aJ/A2 aJ/A2 aJ/A4

force cte. F3,4 f1,2,3 f1,1,2 f1,1,2,2 f1,2,3

description Symmetrized N-H stretch, N-H stretch, N-H stretch, N-H stretch, nondiagonal cubic, cubic, quartic, cubic, nondiagonal nondiagonal nondiagonal nondiagonal value 0.26354 9.0691 0.18856 2.5381 9.0691 2 − − unit aJ/A f1,2,3 f1,1,2 f1,1,2,2 f1,2,3

(6) force cte. f B B1 B2 B3

description mode #2, sextic mode #2 mode#2 mode #2 mode#2 value 2.945 10−5 8272.109 806.32 1163.607 455.543 × unit aJ/A6 cm−1 cm−1 cm−1 cm−1

force cte. D d1 d2 d3 f1,2

description Mode#2 mode#2 mode#2 mode#2 N-Hstretch, nondiagonal value 0.1194 0.36519 0.05925 0.21551 0.09292 − unit dimensionless dimensionless dimensionless dimensionless aJ/A2

force cte. f1,1,2,3 f1,1,1,2 f1,1,2,2,3,3 f1,1,1,2,2,2

description N-H stretch, N-H stretch, N-H stretch, N-H stretch, quartic, quartic, sextic, sextic, nondiagonal nondiagonal nondiagonal nondiagonal value 33.0808 1.7696 2.8084 2.7994 unit aJ/A4 aJ/A4 aJ/A6 aJ/A6

force cte. F1,4,4 F3,4,4 F1,2,2 F3,2,2,4

description cubic cubic, cubic, quartic, symmetrized, symmetrized, symmetrized, symmetrized, nondiagonal nondiagonal nondiagonal nondiagonal value 0.23339 0.14384 0.041673 0.026554 − unit aJ/A3 aJ/A3 aJ/A3 aJ/A4

Table 4 Values for the harmonic and anharmonic force constants, characterizing the ammonia PES, employed in the calculation of the vibrational energy levels in Table 3.