doi:10.1016/S1386-1425(01)00671-0 
Copyright © 2002 Elsevier Science B.V. All rights reserved.
The vibrational levels of ammonia
Céline Léonard
, a, Nicholas C. Handya, Stuart Carterb and Joel M. Bowmanc
a Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
b Department of Chemistry, University of Reading, Reading RG6 2AD, UK
c Cherry L. Emerson Centre for Scientific Computation, Department of Chemistry, Emory University, Atlanta, GA 30322, USA
Received 25 May 2001;
accepted 9 July 2001. ;
Available online 29 November 2001.
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Abstract
A new six-dimensional potential energy function (PEF) of ammonia expressed in internal coordinates is determined by fitting to points evaluated by Density Functional Theory with the B97-1 functional. The C3v and D3h structures are treated on an equal footing. The inversion barrier is 1820 cm−1, which is in very good agreement with the experimental value of 1834 cm−1. The minimum 'reaction path' is well defined by the analytic function up to 40° for the umbrella angle. Using this PEF, the vibrational levels are calculated variationally using three different methods. The first employs the internal kinetic energy operator developed for ammonia by Handy, Carter and Colwell (Mol. Phys. 96 (1999) 477). The second uses the code 
(J. Chem. Phys. 107 (1997) 10458), which involves the kinetic energy operator as expressed in normal coordinates by Watson. The third uses an implementation of the reaction path hamiltonian (J. Chem. Phys. 72 (1980) 99) within the
code. All three approaches give similar energies for the vibrational energies of ammonia, and these agree with experiment to within 15 cm−1 for the fundamental vibrations.
Author Keywords: Internal kinetic energy operator;
; Reaction path hamiltonian
Fig. 1. The internal coordinate system: r1,r2,r3,θ1,θ2,θ3,β.
Fig. 2. Contour plots of sections of the B97-1 PEF for pairs of coordinates defining the six-dimensional expansion. The contour spacing is 10 000 cm−1 and the highest contour is 400 000 cm−1. For each cut, the other coordinates are fixed to their C3v equilibrium values. The angles are in degrees and the distances in bohr.
Fig. 3. Reaction path given by CADPAC (continuous line) and by Eq. (1) (dashed line). The energy values used in this figure are given in Table 4.
Fig. 4. One-dimensional cut along β of the X1A1 and A1A2″ electronic states. r1=r2=r3=1.9 bohr and θ2=θ3=120°.
Fig. 5. Potential along mode Q1, the mode at the inversion saddle point with an imaginary frequency.
Fig. 6. Potential along mode Q4, an NH-stretch without optimisation (‘unoptimised’) and with optimisation (‘optimised’) as described in the text.
Table 1. Stationary points (in Å and degrees), harmonic frequencies (in cm−1) and inversion barrier (in cm−1), for ammonia
Table 2. Grid of points, with their B97-1/TZ2P electronic energies used to represent the PES. r1,r2,r3 are in bohr, θ2,θ3, β are in degrees and energies in hartrees
Table 3. Unique non-zero coefficients (in a.u.) for the PES (Eq. (1))
The other coefficients can be found by recognising that
r1,
r2,
r3 and θ
1,θ
2,θ
3 are equivalent and that (θ
1−(2/3)π)
2=(θ
2−(2/3)π)
2+(θ
3−(2/3)π)
2+2(θ
2−(2/3)π)(θ
3−(2/3)π).
Table 4. The reaction path
The energies are given with respect to the
D3h transition state.
Table 5. Observed [4] J=0 band origins in cm−1 of NH3
Table 6. J=0 band origins in cm−1 of NH3, calculated with the variational code using internal coordinates
Table 7. J=0 band origins in cm−1 of NH3, calculated with MULTIMODE
Table 8. J=0 band origins in cm−1 of NH3, calculated with the reaction path implementation within MULTIMODE
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