Variationally Calculated Vibrational Energy Levels of Ammonia

Feb 23, 2010 - (17) Seabury, C. W.; Rhodin, T. N.; Purtell, R. J.; Merrill, R. P. Surf. Sci. 1980, 93, 117. ... (33) Hedberg, L.; Mills, I. M. J. Mol...
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J. Phys. Chem. C 2010, 114, 4550–4556

Variationally Calculated Vibrational Energy Levels of Ammonia Adsorbed on a Ni(111) Surface Elina Sa¨lli, Vesa Ha¨nninen, and Lauri Halonen* Laboratory of Physical Chemistry, Department of Chemistry, P.O. Box 55 (A.I. Virtasen aukio 1) FI-00014 UniVersity of Helsinki, Finland ReceiVed: NoVember 5, 2009; ReVised Manuscript ReceiVed: December 31, 2009

We have calculated vibrational energy levels for an ammonia molecule adsorbed on a Ni(111) surface. An exact kinetic energy operator for the gas phase molecule is combined with a potential energy function that is calculated with the plane-wave density-functional theory. The resulting eigenvalue problem is solved variationally. The vibrational energy levels for a gas phase molecule are also calculated. The calculated adsorption-related shift is +193 cm-1 for the symmetric bend, -54 cm-1 for the asymmetric bend, -68 cm-1 for the symmetric stretch, and -63 cm-1 for the asymmetric stretch, in good agreement with the experimental values +190, -47, -67, and -84 cm-1, respectively. Introduction Vibrational spectroscopy is an effective tool to fingerprint surface species. Position, intensity, and width of the vibrational bands gives information on the geometry of the molecule and its bonding to the surface. Lateral interactions between adsorbates can be studied by collecting spectral data as a function of the surface coverage. Time resolved spectroscopy makes it possible to monitor surface reactions. Traditionally, spectral assignments of adsorbed molecules have been based on the known frequencies of gas phase molecules or on the normal mode calculations.1 Normal mode calculations for adsorbed species suffer from two distinct sources of error: The harmonic approximation typically overestimates vibrational frequencies, whereas errors in the electronic structure calculations depend on the method. Accurate potential energy surfaces can be calculated for isolated small gas phase molecules, but periodic adsorption systems require the use of less precise density-functional theory. In some systems, with certain exchange-correlation functionals, the errors introduced by the harmonic approximation and electronic structure method may cancel each other. In this paper, we describe the application of the variational method to interpret the high-frequency spectra of an ammonia molecule adsorbed on the Ni(111) surface. The variational method is the exact computational treatment of the vibrational problem within the Born-Oppenheimer approximation if the potential energy surface of the molecule is known and an exact kinetic energy operator is used. The variational calculations have traditionally been used for small gas phase molecules.2 For example, vibrational energy levels of gas phase ammonia have been calculated variationally in refs 3-6. Unfortunately, variational calculations are not feasible for large molecular systems due to the increased number of coordinates involved. However, with the help of an adiabatic assumption, the large systems can often be separated into smaller subsystems where the variational approach can be used. This kind of approach * To whom correspondence [email protected].

should

be

addressed.

E-mail:

has successfully been applied to small water clusters7,8 and adsorption systems.9,10 The present work is a continuation of that work. Adsorption of ammonia on a Ni(111) surface has been studied with several measurement techniques11-20 and computational methods.10,21,22 Ammonia adsorbs on top of a nickel atom like an inverted umbrella with the C3V axis perpendicular to the metal surface. The lone pair electrons of the nitrogen atom bound the molecule to the surface. Vibrational spectra of this adsorption system have been measured with high-resolution electron energy loss spectroscopy (HREELS)11 and infrared reflection absorption spectroscopy (IRAS).12 Upon adsorption, the wavenumber for the symmetric bend blueshifts, whereas wavenumbers for the asymmetric bend and for the stretches redshift.11,12 Theoretical and Computational Methods The Exact Kinetic Energy Operator for an Isolated Ammonia Molecule. A gas phase ammonia molecule has a vibrational double-well potential energy surface associated with the tunneling motion of the nitrogen atom through the plane determined by the hydrogen atoms. The bond lengths r1, r2, and r3 and the bond angles θ1, θ2, and θ3 cannot describe the doublewell potential because the inversion barrier of ammonia is too small to keep the vibrational wave function localized in one of the two potential wells. We decided therefore to use alternative coordinates S2 ) (3-1/4(2π - θ1 - θ2 - θ3)1/2,24 S4a ) 6-1/2(2θ1 - θ2 - θ3), and S4b ) 2-1/2(θ2 - θ3) for the symmetric and degenerate asymmetric bends. The ( signs in the S2 coordinate definition stand for the geometries on the different sides of the planar structure. The value S2 ) 0 corresponds to the planar structure. Within the Born-Oppenheimer approximation, the exact vibrational kinetic energy operator with a unit weight factor in the volume element of integration is25 2

p Tˆ ) - J1/2 2

6

∑

i,j)1

(

)

∂ 1 ∂J (i,j) ∂ -1/2 + g J ∂qi J ∂qi ∂qj

(1)

where qi is r1, r2, r3, S2, S4a, or S4b, the reciprocal metric tensor elements are given by

10.1021/jp9105663  2010 American Chemical Society Published on Web 02/23/2010

Vibrational Energy Levels of Ammonia

J. Phys. Chem. C, Vol. 114, No. 10, 2010 4551 TABLE 1: Results of the Geometry Optimization

4

g(i,j) )

∑ m1R (∇Rqi · ∇Rqj)

(2)

R)1

and J is the Jacobian of the coordinate transformation. The summation index R runs through all atoms in the molecule, and mR is the mass of the Rth nucleus. The g-matrix elements were calculated using eq 2 and the definitions of the S2, S4a, and S4b coordinates as explained in the Supporting Information. The Jacobian of the coordinate transformation is from ref 24. The derivatives dg(i, j)/dqi, dJ/dqi, and d2J/dqidqj were computed by using the chain rule. The kinetic energy operator in eq 1 was arranged to a part that differentiates the wave function 2

p Tˆ′ ) 2

6

∑

i,j)1

(

∂g(i,j) ∂ ∂2 + g(i,j) ∂qi ∂qj ∂qi∂qj

)

(3)

and to a pseudopotential term 2

p Vˆ′ ) 2

6

∑

i,j)1

(

J1/2

∂g(i,j) ∂J-1/2 ∂2J-1/2 + J1/2g(i,j) + ∂qi ∂qj ∂qi∂qj J-1/2g(i,j)

∂J ∂J-1/2 ∂qi ∂qj

)

(4)

Electronic Structure Calculations. All of our electronic structure calculations were based on plane-wave densityfunctional theory as implemented in the Vienna ab initio simulation package (VASP).26,27 We employed the generalized gradient approximation (GGA) by Perdew and Wang (PW91). Calculations were non-spin-polarized. The core electrons were described with the projector augmented wave (PAW) potentials.28 The valence electrons were described by plane-wave basis sets with a cutoff energy 520 eV for the adsorption system and 600 eV for the isolated ammonia molecule. The convergence criteria for the electronic problem was 10-4 eV. Brillouin zone integrations were performed on a nonreduced gamma-centered Monkhorst-Pack grid of 5 × 5 × 1 (13 nonequivalent k-points). The Fermi-level smearing was accomplished with the second order Methfessel-Paxton method using a width of 0.2 eV. The convergence of the results was tested against the size of the basis set. The effect of spin polarization was studied by Kurten et al.10 They found out that the effect is small for this system even though nickel is a magnetic material. The metal surface was modeled with a slab model. A lattice constant of 3.515 Å was obtained by computing the total energy of a bulk nickel super cell of 27 atoms and fitting the data to an analytic function. The super cell contained five layers of nickel atoms with nine atoms in each layer and a vacuum of 10 Å to isolate slabs from their periodic images. The size of the super cell was chosen to mimic the strong repulsive interactions between adjacent ammonia molecules. The isolated ammonia molecule was put to a large cubic super cell that had dimensions of 15 Å. Geometry Optimization. The relaxed geometry of the adsorption system was needed to fix the translational and rotational coordinates of the system. Instead of performing an extensive study of the possible adsorption sites and geometries, the geometry where the ammonia molecule is on top of a nickel atom with the C3V axis perpendicular to the metal surface was used as a starting point in the geometry optimization. As the

rNH (Å) θHNH (deg) rNNi (Å) ∆rNiNi (Å) Eads (eV)

isolated NH3

NH3 on Ni(111)

1.021 106.68

1.023 108.67 1.99 0.21 -0.91

Ni(111) surface plane is hexagonal, the only coordinate that determines whether the adsorption geometry is symmetrical is the rotational angle around the C3V axes. We calculated a rotational barrier of 0.001 eV around the C3V axis, which is in good agreement with electron stimulated desorption ion angular distribution (ESDIAD) measurements13-15 and previous computational works.10,21,22 As the molecule essentially rotates freely on the surface, we decided to maintain the gas phase symmetry of the molecule by aligning the N-H bonds of the adsorbed molecule with the Ni rows. This facilitates the time-consuming plane-wave calculations. The metal surface without ammonia, the isolated molecule, and the adsorption system were relaxed until the HellmannFeynman forces of all unconstrained atoms were less than 0.001 eV Å-1. In the relaxations, the two uppermost layers were allowed to relax and the three bottom layers were kept fixed. The results of the geometry optimization are shown in Table 1. Ammonia adsorbs on top of a nickel atom through the nitrogen atom with the C3V axis parallel to the surface normal. The N-H vectors are aligned with the Ni rows. The distance between the nitrogen atom and the nickel atom below it is 1.99 Å. The nickel atom right below the molecule shifts up by 0.21 Å ()∆rNiNi). The ammonia molecule maintains its gas phase symmetry, and its bond angles and bond lengths increase only a little. Our calculated geometry values are in good agreement with earlier computational results.10,21,22 The calculated distance 1.99 Å from the nitrogen atom to the closest nickel atom agrees well with a value of 1.97 Å determined from photoelectron diffraction measurements.18 The computed adsorption energy, Eads ) ENH3+surface - ENH3 - Esurface, is -0.91 eV. An experimental estimate of the adsorption energy can be obtained from the temperature resolved desorption experiments. The desorption temperatures for an ammonia layer chemisorbed on the Ni(111) surface vary from 120 to 350 K with peak maxima from 270 to 300 K.11,14-17 The large temperature range and the shape of the desorption features indicate strong repulsive interactions between adjacent ammonia molecules. In reasonable agreement with our ab initio value, we have calculated values from -0.83 to -0.74 eV for the adsorption energy by using the Redhead formula29 (a value 1013 K-1 has been used for the quotient of the frequency factor and the heating rate). Definition of the Vibrational Coordinates for the Adsorption System. The stretching parts of potential energy surfaces were expressed in symmetrized bond length coordinates S1 ) 3-1/2(∆r1 + ∆r2 + ∆r3), S3a ) 6-1/2(2∆r1 - ∆r2 - ∆r3), and S3b ) 2-1/2(∆r2 - ∆r3), where ∆ri is the displacement from the equilibrium bond length [the coordinates are defined so that θ1 ) ∠(r2, r3), θ2 ) ∠(r3, r1), and θ3 ) ∠(r1, r2)]. The molecule vibrates around a fixed translational and rotational equilibrium geometry defined by the following conditions: (i) The structure of the metal surface is fixed. (ii) The translational coordinates of the ammonia molecule are defined by fixing the position of the nitrogen atom. The center of mass of the molecule would be another

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Sa¨lli et al. The S2 coordinate has a nonzero equilibrium geometry value unlike the other coordinates. Thus, the lower boundary for the summation index k is 0 for the symmetric bend resulting in a constant term, which is the height of the inversion barrier and 2 for all other coordinates. Two-dimensional data were fitted to the potential energy function

Vi,j(qi, qj) ) Vi(qi) + Vj(qj) +

∑ Fq q qikqjl k,l

k l i j

(7)

when qi, qj * S2 or

Figure 1. The symmetry plane used to define the coordinates S3a and S4a for the adsorbed ammonia molecule. The bond length ri is associated with the bond NHi. The hydrogen atom H1 is on the symmetry plane. The positions of the two hydrogen atoms H2 and H3 are identical regardless of the coordinate values S1, S2, S3a, and S4a.

choice, but it would make the potential energy surface mass-dependent, which is undesirable. (iii) The tilting angles of the molecule were fixed by keeping the angles between the different N-H vectors and the normal of the metal surface equal to each other. (iv) The symmetry of the adsorption system was used to fix the rotational motion around the C3V axes. (a) For the symmetric stretch and bend, we kept the projections of the N-H vectors on the metal surface aligned with the Ni rows. As the adsorbed ammonia is a nearly free rotor, we believe that this assumption has only a little effect on the potential energy surfaces. (b) For the asymmetric stretch and bend, we maintained the symmetry plane shown in Figure 1. Because of the C3V symmetry of the system, we could use symmetry relations to obtain all relevant potential energy terms containing the S4b coordinate without defining the fixed translational and rotational equilibrium geometry for it. The S3b coordinate was defined so that all of the N-H-Ni angles were fixed. Calculation of the Potential Energy Surfaces. Each potential energy surface was calculated at discrete data points on the one-dimensional S1, S2, S3a, S3b, and S4a surfaces and on the two-dimensional S1S2, S1S3a, S1S4a, S2S3a, S2S4a, and S3aS4a surfaces. The effect of the potential energy parameters obtained from the three-dimensional S1S2S3a, S1S2S4a, S1S3aS4a, and S1S3aS4a surfaces was tested, and the results will be discussed later. When we computed each of the surfaces, we set the other coordinates to their equilibrium values. The S2 coordinate for the gas phase ammonia was set to one of the two possible nonzero equilibrium values. Data points were fitted to analytical potential energy functions by using the nonlinear least-squares method. Symmetric stretching data were fitted to Morse coordinate power series

V1(S1) )

∑ Dk(1 - e-aS )k 1

(5)

k

Other one-dimensional data sets were fitted to power series

Vi(qi) )

∑ Fq qik k

k i

(6)

V2,j(S2, qj) ) V2(S2) + Vj(qj) - V2(S2(eq)) +

∑ FS q [Sk2 - S2(eq)k]qjl k,l

k l 2 j

(8)

where the nonzero equilibrium value of the S2 coordinate in all calculated one-dimensional surfaces (except naturally in the S2 surface) was taken into account. This kind of procedure is necessary because the calculated Si surfaces, other than S2, include nonzero two-dimensional potential energy terms where the S2 coordinate is constrained to the equilibrium value, and we do not want to doublecount them. Symmetry relations were used to obtain the parameters for the one-dimensional S3b and S4b surfaces, and for the other parameters obtained from the calculated surfaces. The S3b surfaces were computed in order to obtain the coefficient of 6 terms. the S3b Variational Calculations. The eigenvalue problems of the Hamiltonians were solved variationally in six-dimensional basis sets that were constructed as products of onedimensional Morse oscillator eigenfunctions for the stretches and one-dimensional harmonic oscillator basis functions for the bends. Analytical formulas for Morse oscillator eigenfunctions were take from ref 30. All integrals were separated into stretching and bending parts. The stretching parts were computed with three-dimensional numerical Gauss-Laguerre integration and the bending parts with three-dimensional numerical Gauss-Hermite integration. Basis set contraction was used to reduce the number of basis functions needed in the final six-dimensional calculation and to avoid the problems associated with the coupled integration limits for the bending coordinates.6 The idea is to solve the vibrational problems of the lower-dimensional systems and use the eigenfunctions obtained as basis functions in the final sixdimensional calculation. See, for example, ref 31 for the use of basis set contraction. A schematic picture of the procedure can be seen in Figure 2. A one-dimensional subproblem of the symmetric bend (step 1) and a two-dimensional subproblem of asymmetric bends (step 2) were solved to obtain better basis sets for symmetric and asymmetric bends. Then, the threedimensional subproblem of the bends was solved (step 3), producing an improved basis set for the bends. A threedimensional subproblem of the stretches was solved (step 4) to obtain an improved basis set for the stretches. The threedimensional stretching and bending wave functions obtained in this way were finally used in the six-dimensional variational calculation (step 5). The exact kinetic energy operator Tˆ ) Tˆ′ + Vˆ′ was used in the six-dimensional calculation, but the reduced kinetic energy operator Tˆ′ was used in the basis set contraction steps 1-4. Only the coordinates that were included were taken into account in Tˆ′ and Vˆ for each of the basis set

Vibrational Energy Levels of Ammonia

J. Phys. Chem. C, Vol. 114, No. 10, 2010 4553 TABLE 3: The Coordinate Values for the Data Points on the Two-Dimensional Potential Energy Gridsa coordinate

data points

S1 S2 S3a S4a

(0.17 or (0.087 Å 0.20, 0.40, 0.53, or 0.72 rad1/2 (0.12 or (0.25 Å (0.17 or (0.34 rad

a Additional data points, not listed here, were needed for the S1S2, S2S3a, and S2S4a surfaces due to the complex nature of the double-well potential for the gas phase molecule and the highly repulsive part of the S2-potential energy curve of the adsorption system.

Figure 2. The five basis set contraction steps described in the text were implemented in five variational programs of which the first four produce basis functions for the final program.

TABLE 2: The Numbers and Ranges for the Calculated One-Dimensional Data Points of the Gas Phase Moleculea coordinate

number of data points

S1 S2 S3a S3b S4a

39 50 26 20 41

range of coordinates from from from from from

-0.26 to 0.52 Å 0 to 1.27 rad1/2 -0.24 to 0.37 Å -0.28 to 0.28 Å -0.94 to 0.94 rad

a In the adsorption system, the ranges of the coordinates were similar but points with negative values of the S2 coordinate were also calculated. The numbers of data points were at least half of the number of data points for the gas phase system.

contraction steps. All other coordinates were set to their equilibrium values. For the symmetric bend, asymmetric bend, and stretching subproblems, the potential energy surfaces were obtained from the fittings using eqs 5-8. For the bending subproblem and for the final calculation with six coordinates, the one-dimensional stretching and asymmetric bending potential energy functions (qi) were obtained as Vfinal i

Vifinal(qi) ) Vi(qi) -

∑ FS q S2(eq)kqil k,l

k l 2 i

(9)

because the calculated one-dimensional asymmetric bend and stretching surfaces include the two-dimensional potential energy terms with the S2 coordinate set to its nonzero equilibrium value. The variational method described was implemented in a computer program which was tested using the vibrational potential energy function of ammonia by Martin et al.32 Our computed vibrational wavenumbers for the fundamental transitions agree within 0.03 cm-1 with the values from refs 3 and 5. Results and Discussion Results of Potential Energy Surface Calculations. The calculated electronic energy data points that are shown in Tables 2 and 3 were fitted to the potential energy functions as explained earlier. The potential energy coefficients obtained are shown in Table 4. Parameters that can be obtained from the given parameters using symmetry relations have been excluded from the table. See ref 32 for more details. As examples, the symmetric stretching and symmetric bending potential energy curves are shown in Figures 3 and 4. The

bond lengths of the ammonia molecule increase and the dissociation energy corresponding to the symmetric stretch decreases in adsorption. This implies weakening of the N-H bonds and red-shifted vibrational wavenumbers for the symmetric stretch. Figure 4 shows that the bond angles of ammonia increase when it adsorbs on a surface. The symmetric bending potential of the adsorbed molecule has only one minimum. The constant term in the symmetric bending potential energy surface that describes the energy difference between the planar structure and the equilibrium geometry increases from 0.22 to 0.48 eV upon adsorption. This explains the blueshifted vibrational frequencies for the symmetric bend that we report below. Results of the Variational Calculations. For the symmetric bend problem, 80 basis functions and 90 integration points were used. All quantum numbers associated with the S4a and S4b coordinates up to the sum 17 were used in the asymmetric bend problem, resulting in 153 two-dimensional wave functions with 20 integration points for both asymmetric bend coordinates. In the three-dimensional bending calculation, 12 of the symmetric bend wave functions and 45 of the asymmetric bend wave functions were used with 36 integration points for the symmetric bend coordinate and 16 integration points for both asymmetric bend coordinates. All quantum numbers for the stretching coordinates up to the sum 10 were used in the stretching problem, resulting in 220 three-dimensional wave functions. The quadrature grid for the stretches was made from 30 integration points for one bond length. The eigenvalue problem of the complete Hamiltonian was finally solved by using 98 contracted stretching wave functions (91 for the adsorbed molecule) and 121 contracted bending wave functions (120 for the adsorbed molecule). Alltogether, 20 integration points were used for each of the stretches, 30 integration points were used for the symmetric bend, and 20 integration points were used for each of the asymmetric bend coordinates. For comparison, we also performed normal mode calculations for gas phase and adsorbed ammonia molecule using the computer program ASYM20.33 The equilibrium geometries were obtained from the geometry relaxations, and the harmonic force constants were calculated as second derivatives of our potential energy functions. The ordinary symmetry coordinate 3-1/2(∆θ1 + ∆θ2 + ∆θ3) was used for the symmetric bend. The results of the anharmonic variational calculations together with calculated harmonic and experimental wavenumbers are shown in Table 5. Only four fundamental wavenumbers are reported because the asymmetric bend and asymmetric stretch are degenerate. The variationally calculated and experimental vibrational energy levels of the gas phase molecule are split to the symmetric (+) and asymmetric (-) component because of the inversion. The harmonic energy levels of the gas phase molecule are not split because the harmonic approximation does not allow the molecule to invert. The calculated anharmonic

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Sa¨lli et al.

TABLE 4: Potential Energy Surface Parameters parameter a D2 D3 D4 FS02 FS2 FS22 FS23 FS42 FS52 FS62 FS72 FS28 FS122 FS23a FS33a FS43a FS3a5 FS63a FS63b FS24a FS34a FS4a4 FS54a FS1S23a FS1S33a FS1S43a FS1S3a5 FS21S23a FS21S33a FS21S43a FS31S23a FS13S3a3

unit -1

Å eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV

gas 1.264 12.98 1.216 0.2206 -1.041 1.076 0.4052

Å-2 Å-3 Å-4 Å-5 Å-6 Å-6

Å-3 Å-4 Å-5 Å-6 Å-4 Å-5 Å-6 Å-5 Å-6

-0.09073 0.01767 21.02 -18.29 35.84 -36.41 30.03 27.12 1.979 -0.07627 0.06465 -0.02240 -77.21 64.81 -159.6 150.2 138.7 -152.5 305.8 -212.0 204.2

ads 1.392 10.49 0.8888 1.034 0.4794 -1.100 0.02165 0.08536 1.082 0.05470 0.3546 -0.33230 0.1823 20.41 -18.25 35.28 -35.28 27.85 33.60 1.813 -0.1129 0.04466 -0.007237 -76.87 64.27 -157.2 140.7 136.6 -150.6 290.7 -210.4 197.2

parameter FS41S23a FS2S24a FS2S4a3 FS22S24a FS22S34a FS22S44a FS32S24a FS24S4a2 FS42S34a FS1S2 FS1S22 FS1S32 FS1S24 FS1S62 FS21S2 FS21S22 FS21S42 FS1S4a2 FS1S34a FS1S44a FS21S24a FS2S23a FS2S3a3 FS22S23a FS22S33a FS42S33a FS3aS4a FS3a2 S4a FS3aS24a FS33aS4a FS23aS24a FS3aS34a

unit eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV eV

Å

-6

gas 195.2 1.695 0.5112 0.5530

ads 212.3 0.2373 0.07129 1.474 0.2687 -0.1260

-0.1041 -0.1026 Å-1 Å-1 Å-1 Å-1 Å-1 Å-2 Å-2 Å-2 Å-1 Å-1 Å-1 Å-2 Å-2 Å-3 Å-2 Å-3 Å-3 Å-1 Å-2 Å-1 Å-3 Å-2 Å-1

-3.594

1.194 -2.589 -0.4399

0.8760 -0.5687 -0.7991 1.611 -0.9774 0.2128 0.7867 0.2825 -0.1296 0.1442 0.4369 -1.164 -1.134 0.1801 0.4889 -0.2793 -0.3727

1.195 0.4157 -1.135

1.466 0.7539 -1.484 -0.9824 -1.086 0.2519 0.5340 -0.4232 -0.2586

and experimental energy levels of the adsorbed molecule are not split because the inversion barrier is high, and therefore, the wave function is localized in the potential energy well where the hydrogen atoms point away from the metal surface. Convergence of the calculated wavenumbers was tested for the gas phase ammonia. Convergence with respect to the basis set is better than 0.17 cm-1 for all reported transitions. The results are also converged with respect to the number of integration points. The convergence with respect to the expansion of potential energy function was tested by calculating additional data points on the three-dimensional S1S2S3a, S1S2S4a, S1S3aS4a, and S1S3aS4a surfaces and following the procedure similar to one explained earlier. The results of the variational

calculations using the extended potential are shown in parentheses after each calculated gas phase value in Table 5. The effect of the additional potential energy terms is less than 2.5 cm-1 for the bends and asymmetric stretch, and the inversion splittings are essentially unaffected. The three-dimensional terms are more important for the asymmetric stretch where the symmetric level (+) increases by 5 cm-1 and the inversion splitting for the asymmetric stretch decreases from 4.8 to 1.0 cm-1, which is much closer to the experimental value 0.4 cm-1 than the value from the potential energy model without the threedimensional potential energy terms. It is likely that the additional high-order terms have a similar small effect on the adsorption system and thus do not significantly change adsorption-related

Figure 3. The potential energy curves for the symmetric stretch.

Figure 4. The potential energy curves for the symmetric bend.

Vibrational Energy Levels of Ammonia

J. Phys. Chem. C, Vol. 114, No. 10, 2010 4555

TABLE 5: Calculated Anharmonic Variational, Calculated Harmonic, and Experimental Vibrational Wavenumbers (cm-1) for the Symmetric Bend ν2, Degenerate Asymmetric Bend ν4, Symmetric Stretch ν1, and Degenerate Asymmetric Stretch ν3a gas, anharmb GS splitting ν2 + ν4 + + ν1 + ν3 -

1.0 (1.0) 874.6 (876.2) 919.0 (919.8) 1572.9 (1573.7) 1574.5 (1575.2) 3250.9 (3253.4) 3252.5 (3254.9) 3350.7 (3355.7) 3355.5 (3356.7)

gas, harmb gas, expc ads, anharmb ads, harmb 1000 1616 3391 3514

0.8 932.4 968.1 1626.3 1627.3 3336.1 3337.1 3443.6 3444.0

ads, exp

anharm shiftb harm shiftb

exp shift

1090

1151

1140,d 1113e

+193

+151

+190,d +163e

1520

1569

1580d

-54

-47

-47d

3184

3350

3270,d 3251e

-68

-41

-67,d -86e

3290

3466

3360d

-63

-48

-84d

a The plus and minus signs in the second column stand for the energy levels of the gas phase ammonia molecule that are split because of the inversion. Values in parentheses are obtained by adding potential energy terms obtained from the three-dimensional potential energy surfaces to the variational calculation. See text for more details. Shifts are calculated by using the average of the gas phase inversion doublet values. The calculated ground state (GS) energy is 7209.7 cm-1 for the gas phase molecule (7210.8 cm-1 if additional potential energy terms are included) and 7158 cm-1 for the adsorbed molecule. b This work. c Reference 34. d Reference 11. e Reference 12.

shifts in the vibrational wavenumbers. Therefore, due to the large computational cost of the electronic structure calculations, we decided to not calculate these additional potential energy terms for the adsorption system. The experimental wavenumbers for the gas phase ammonia are taken from ref 34. The experimental wavenumbers for the adsorbed molecule are measured with high-resolution electron energy loss spectroscopy (HREELS)11 or Fourier transformation infrared reflection absorption spectroscopy (FT-IRAS).12 Asymmetric modes are forbidden in the IRAS spectra because of the strong surface selection rule. In the FT-IRAS experiment, the coverage of the surface was 0.14 NH3/Ni which is close to our computational coverage 0.11 NH3/Ni. The HREEL spectrum is from first layer ammonia molecules chemisorbed on the surface. By comparing the HREEL spectrum with the low-coverage spectra of ammonia adsorbed on a Pt(111) surface from ref 23, we believe that it also corresponds to a low coverage case. However, it is unclear which of the experiments corresponds closer to our coverage. Our calculated vibrational wavenumbers are systematically 2-6% lower than the experimental values. This kind of systematic deviation is due to inaccuracies of the densityfunctional theory. In order to better compare a gas phase and adsorbed ammonia molecule, we calculated shifts in the vibrational wavenumbers from the gas phase values to the adsorbed molecule values. The wavenumber for the isolated molecule is calculated as an average of the ( levels. The results are shown in Table 5. The most significant change is the blueshift 193 cm-1 of the symmetric bend due to the more repulsive symmetric bend potential. The wavenumbers of the asymmetric bend, symmetric stretch, and asymmetric stretch redshift by 54, 68, and 63 cm-1, respectively. As bonding to the surface typically makes the internal structure of the molecule looser, these redshifts were expected. The anharmonic shifts are in good agreement with the experimental shifts, especially when they are compared with the values from ref 11. Adsorption -related shifts in the harmonic wavenumbers are similar to the anharmonic ones, but for the stretches, the anharmonic calculations give clearly better agreement with experimental shifts than the normal-mode analysis. Conclusions We have calculated vibrational energy levels for the gas phase ammonia molecule and the ammonia molecule adsorbed on a Ni(111) surface. Our work is based on the assumption that the high-frequency modes of ammonia are not coupled to the

rotational and translational modes of the molecule with respect to the surface. Plane-wave density-functional theory has been used to calculate the potential energy surfaces. An exact kinetic energy operator of an isolated molecule has been employed for both systems. The resulting eigenvalue equations have been solved variationally. Our vibrational model is anharmonic and takes couplings between different modes into account. Especially, it enables us to calculate inversion splittings and highly anharmonic vibrational energy levels for the gas phase ammonia molecule. Upon adsorption, the wavenumber for the symmetric bend blueshifts, whereas wavenumbers for the asymmetric bend and for the stretches redshift. Our calculated shifts are in good agreement with the corresponding experimental values. The part of the error introduced by the density-functional theory potential energy surfaces is canceled when adsorption-related shifts are calculated. Acknowledgment. We thank CSC Scientific Computing Ltd for providing the computational time. E.S. thanks the Finnish Cultural Foundation and Magnus Ehnrooth Foundation for the financial support. We thank the Academy of Finland for funding the Finnish Center of Excellence in Computational Molecular Science. Supporting Information Available: Information on how the g-matrix elements were calculated and the analytical expression for the Jacobian of the coordinate transformation. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Mills, P.; Jentz, D.; Trenary, M. J. Mol. Catal. A 1998, 131, 209. (2) Jensen, P.; Bunker, P. R. Computational molecular spectroscopy; John Wiley & Sons Ltd: England, 2000. (3) Handy, N. C.; Carter, S.; Colwell, S. M. Mol. Phys. 1999, 96, 477. (4) Gatti, F.; Lung, C.; Leforestier, C.; Chapuisat, X. J. Chem. Phys. 1999, 111, 7236. (5) Luckhaus, D. J. Chem. Phys. 2000, 113, 1329. (6) Rajama¨ki, T.; Miani, A.; Halonen, L. J. Chem. Phys. 2003, 118, 6358. (7) Salmi, T.; Ha¨nninen, V.; Garden, A. L.; Kjaergaard, H. G.; Tennyson, J.; Halonen, L. J. Phys. Chem. A 2008, 112, 6305. (8) Salmi, T.; Kjaergaard, H. G.; Halonen, L. J. Phys. Chem. A 2009, 113, 9124. (9) Sa¨lli, E.; Jalkanen, J.-P.; Laasonen, K.; Halonen, L. Mol. Phys. 2007, 105, 1271. (10) Kurten, T.; Biczysko, M.; Rajama¨ki, T.; Laasonen, K.; Halonen, L. J. Phys. Chem. B 2005, 109, 8954.

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