Comparison of methods for solving the vibrational Schrödinger equation in the course of sequential Monte-Carlo-quantum mechanical treatment of hydroxide ion hydration


  • Jasmina Petreska Institute of Chemistry, Faculty of Natural Sciences and Methematics, Ss. Cyril and Methodius University, Arhimedova 5, P.O. Box 162, 1000 Skopje
  • Ljupco Pejov Institute of Chemistry, Faculty of Natural Sciences and Methematics, Ss. Cyril and Methodius University, Arhimedova 5, P.O. Box 162, 1000 Skopje



hydroxide ion, ionic water solutions, solvation, hydrogen bonds, intermolecular interactions, anharmonic O–H vibrational frequency shifts, Monte-Carlo simulation, Fourier grid, Hamiltonian method, Numerov algorithm, diagonalization of Hamiltonian matrix


Three numerical methods were applied to compute the anharmonic O–H stretching vibrational frequencies of the free and aqueous hydroxide ion on the basis of one-dimensional vibrational potential energies computed at various levels of theory: i) simple Hamiltonian matrix diagonalization technique, based on representation of the vibrational potential in Simons-Parr-Finlan (SPF) coordinates, ii) Numerov algorithm and iii) Fourier grid Hamiltonian method (FGH).
Considering the Numerov algorithm as a reference method, the diagonalization technique performs remarkably well in a very wide range of frequencies and frequency shifts (up to 300 cm–1). FGH method, on the other hand, though showing a very good performance as well, exhibits more significant (and non-uniform) discrepancies with the Numerov algorithm, even for rather modest frequency shifts.


V. Kocevski, Lj. Pejov, On the assessment of some new meta-hybrid and generalized gradient approximation functionals for calculations of anharmonic vibrational fre-quency shifts in hydrogen-bonded dimers. J. Phys. Chem. A, 114, 4354-4363 (2010).

Lj. Pejov, K. Hermansson, On the nature of blueshifting hydrogen bonds: ab initio and density functional studies of several fluoroform complexes. J. Chem. Phys., 119, 313-324 (2003).

Lj. Pejov, A gradient-corrected density functional and MP2 study of phenol-ammonia and phenol-ammonia(+) hydrogen bonded complexes. Chem. Phys., 285, 177-193 (2002).

B. Silvi, R. Wieczorek, Z. Latajka, M. E. Alikhani, A. Dkhissi, Y. Bouteiller, Critical analysis of the calculated frequency shifts of hydrogen-bonded complexes, J. Chem. Phys., 111, 6671-6678 (1999).

C. E. Blom, C. Altona, Application of self-consistent-field ab-initio calculations to organic-molecules. 2. Scale factor method for calculation of vibrational frequencies from ab-initio force constants - ethane, propane and cyclopropane, Mol. Phys., 31, 1377-1391 (1976).

P. Pulay, G. Fogarasi, G. Pongor, J. E. Boggs, A. Vargha, Combination of theoretical ab initio and experimental information to obtain reliable harmonic force-constants - scaled quantum-mechanical (SQM) force-fields for glyoxal, acrolein, butadiene, for-maldehyde, and ethylene, J. Am. Chem. Soc., 105, 7037-7047 (1983).

G. Fogarasi, Recent developments in the method of SQM force fields with application to 1-methyladenine, Spectrochim. Acta A, 53, 1211-1224 (1997).

G. Szasz, A. Kovacs, Investigation of the density-functional theory-derived scaled quantum mechanical method for cage-like systems: the vibrational analysis of adamantine, Mol. Phys., 96, 161-167 (1996).

A. Kovacs, V. Izvekov, G. Keresztury, G. Pongor, Vibrational analysis of 2-nitrophenol. A joint FT-IR, FT-Raman and scaled quantum mechanical study, Chem. Phys., 238, 231-243 (1998).

R. B. Gerber, J. O. Jung, in Computational Molecular Spectroscopy, P. Jensen, P. R. Bunker (eds.), Wiley and Sons: Chichester, 2000.

G. M. Chaban, J. O. Jung, R. B. Gerber, Ab initio calculation of anharmonic vibra-tional states of polyatomic systems: Electronic structure combined with vibrational self-consistent field, J. Chem. Phys., 111, 1823-1829 (1999).

V. Barone, Anharmonic vibrational properties by a fully automated second-order perturbative approach, J. Chem. Phys., 122, 14108 (2005).

F. Gangemi, R. Gangemi, G. Longhi, S. Abbate, Calculations of overtone NIR and NIR-VCD spectra in the local mode approximation: camphor and camphorquinone, Vib. Spectrosc., 50, 257-267 (2009).

F. Gangemi, R. Gangemi, G. Longhi, S. Abbate, Experimental and ab initio calculated VCD spectra of the first OH-stretching overtone of (1R)-(-) and (1S)-(+)-endo-borneol, Phys. Chem. Chem. Phys., 11, 2683-2689 (2009).

G. Simons, R. G. Parr, J. M. Finlan, New alternative to dunham potential for diatomic-molecules, J. Chem. Phys., 59, 3229-3234 (1973).

C. C. Marston, G. G. Balint-Kurti, The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions, J. Chem. Phys., 91, 3571-3576 (1989).

R. D. Johnson, FGHD1 – program for one-dimensional solution of the Schrödinger equation, Version 1.01.

R. J. Le Roy, LEVEL 8.0: A Computer Program for Solving the Radial Schröodinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663 (2007); see

J. Petreska, Lj. Pejov, K. Coutinho, K. Hermansson, submitted for publication.

T. Corridoni, A. Sodo, F. Bruni, M. A. Ricci, M. Nardone, Probing water dynamics with OH-, Chem. Phys., 336, 183-187 (2007).

H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, J. Hermans, Intermolecular Forces, edited by B. Pullmam (Reidel, Dordrecht, 1981), p.331.

I. S. Ufimtsev, A. G. Kalinichev, T. J. Martinez, and R. James Kirkpatrick, Chem. Phys. Lett. 442, 128 (2007).

K. Coutinho, M. J. de Oliveira, S. Canuto, Sampling configurations in Monte Carlo simulations for quantum mechanical studies of solvent effects, Int. J. Quantum Chem., 66, 249-253 (1998).

K. Coutinho, S. Canuto, Solvent effects in emission spectroscopy: A Monte Carlo quantum mechanics study of the n <-pi(*) shift of formaldehyde in water, J. Chem. Phys., 113, 9132-9139 (2000).

S. Canuto, K. Coutinho, Solvent effects from a sequential Monte Carlo - Quantum mechanical approach, Adv. Quantum Chem., 28, 89-105 (1997).

K. Coutinho, S. Canuto, M. C. Zerner, A Monte Carlo-quantum mechanics study of the solvatochromic shifts of the lowest transition of benzene, J. Chem. Phys., 112, 9874-9880 (2000).

C. Chatfield, The Analysis of Time Series. An Introduction, Chapman and Hall, Lon-don (1984).

R. Krätschmer, K. Binder, D.Stauffer, Linear and nonlinear relaxation and cluster dynamics near critical points, J. Stat. Phys. 15, 267-297 (1976).

A. D. Becke, Density-functional thermochemistry. 3. The role of exact exchange, J. Chem. Phys., 98, 5648-5652 (1993).

C. Lee, W. Yang, R. G. Parr, Development of the colle-salvetti correlation-energy formula into a functional of the electron-density, Phys. Rev. B, 37, 785-789 (1988).

K. Coutinho, S. Canuto, DICE: A Monte Carlo program for molecular liquid simulation, University of São Paulo, Brazil, 1997.

M. J. Frisch et al, Gaussian 03, Revision C.02, Gaussian, Inc. Wallingford CT, 2004.

A. Shayesteh, R. D. E. Henderson, R. J. Le Roy, P. F. Bernath, Ground state potential energy curve and dissociation energy of MgH, J. Phys. Chem. A, 111, 12495-12505 (2007).

H. Li, R. J. Le Roy, Spectroscopic properties of MgH2, MgD2, and MgHD calculated from a new ab initio potential energy surface, J. Phys. Chem. A, 111, 6248-6255 (2007).

J. C. Owrutsky, N. H. Rosenbaum, L. M. Tack, R. J. Saykally, The vibration-rotation spectrum of the hydroxide anion (OH-), J. Chem. Phys., 83, 5338-5339 (1985).




How to Cite

Petreska, J., & Pejov, L. (2010). Comparison of methods for solving the vibrational Schrödinger equation in the course of sequential Monte-Carlo-quantum mechanical treatment of hydroxide ion hydration. Macedonian Journal of Chemistry and Chemical Engineering, 29(2), 203–213.



Theoretical Chemistry