Toward first-principles complete spectroscopy of small molecules

Transcription

1 Theses of the Ph. D. Dissertation GÁBOR CZAKÓ Toward first-principles complete spectroscopy of small molecules Ph. D. Advisers: Prof. Dr. Attila G. Császár Laboratory of Molecular Spectroscopy, Institute of Chemistry, Eötvös University Dr. Viktor Szalay Crystal Physics Laboratory, Research Institute for Solid State Physics and Optics, ungarian Academy of Sciences Chemistry Doctoral School ead of Doctoral School: Prof. Dr. György Inzelt Theoretical and Physical Chemistry and Material Science Program ead of Doctoral Program: Prof. Dr. Péter R. Surján Eötvös Loránd University Budapest, 2007

2 I. INTRODUCTION Experimental determination of all the rotational-vibrational transitions of a molecule is impossible due to the huge number of possible transitions, on the order of billions even for a triatomic molecule. Furthermore, the measurements are often exceedingly difficult; for example, in the case of radicals due to their short lifetimes. Thus, only theory is able to provide the complete rovibrational spectrum of a molecule. One possibility for the numerical solution of the (ro)vibrational Schrödinger equation is based on perturbation theory. Vibrational perturbation theory carried out to second order (VPT2) is the traditional route. Within VPT2 the anharmonic vibrational frequencies can be obtained using analytical formulae from a quartic force field expressed in normal coordinates. It is important to note that, unlike the harmonic vibrational analysis, VPT2 provides reasonable estimates not only for the fundamentals but also for vibrational overtones and combination bands. owever, VPT2 includes severe approximations. If one wants to perform high-accuracy first-principles computations or the determination of the complete spectrum is the goal, one should employ the variational technique for solving the nuclear Schrödinger equation. The converged variational results are the exact rovibrational energy levels corresponding to the given potential energy surface (PES). The principal goal of my doctoral research was to develop variational techniques in order to compute (ro)vibrational states of triatomic molecules with high accuracy. I was also interested in applying my newly developed methods for computing (ro)vibrational energy levels and (ro)vibrationally averaged properties of different molecules. Traditionally, computation of vibrationally averaged molecular properties, e.g., average distances or effective rotational constants, involves a harmonic or anharmonic vibrational analysis and normal coordinates. I wanted to follow a different route, namely the variational computation of rovibrationally averaged properties. I focused on studying the 2 O molecule and the molecular ion. Arguably water is the most important polyatomic molecule; for example, it plays a vital role in the greenhouse effect on Earth. Therefore, determination of the complete rotational-vibratrional spectrum of water is a requirement in the quantitative understanding of the greenhouse effect. Determination of the complete (ro)vibrational spectrum of, one of the most important systems in astrophysics, is an exceedingly challenging task for theory. Although singularities are always present in amiltonians given in an internal coordinate representation, treatment of the singularities is essential only if the singular point is accessible 2

3 in the energy region of interest. Due to the fact that the complete (ro)vibrational spectrum of cannot be computed in orthogonal internal coordinates without treating the important singularities, I was working on developing different techniques for solving the singularity problem in triatomics. In the past two decades grid techniques, in particular the discrete variable representation (DVR) methods, became popular in (ro)vibrational computations. The DVR via the so-called transformation method was introduced in 1965 by arris and co-workers. In 1968 Dickinson and Certain showed its close relation to Gaussian quadratures, which explained its high accuracy in the case of a polynomial basis. Before I started my doctoral research, we had extended the proof of Dickinson and Certain and showed that the transformation method gives matrix elements of Gaussian quadrature accuracy even in the case of one-dimensional general bases. During my doctoral research I continued studying the different related techniques, which can be employed for determining the matrix representation of the (ro)vibrational amiltonians. The variational (ro)vibrational calculations are based usually on the so-called Born Oppenheimer (BO) approximation. The BO approximation separates the motion of the nuclei from that of the electrons and allows dividing quantum chemistry into two main parts; namely, electronic structure and nuclear motion calculations. If one is set to achieve true spectroscopic accuracy in the computations, which defined to be 1 cm 1 on average by quantum chemists, one has to move beyond the BO approximation. One can try to solve the Schrödinger equation based on an atomic or a molecular amiltonian involving operators that depend on all the electronic and all the nuclear coordinates. This treatment is the so-called nonadiabatic approach, which provides the exact nonrelativistic states of a system with a Coulombic potential. In electronic structure theory another approach, namely that based on the diagonal BO correction (DBOC) has been employed to go beyond the BO approximation. The DBOC is a first-order perturbative correction of the electronic energy, which is the average value of the nuclear kinetic energy operator relative to the BO electronic wave function. This so-called adiabatic correction, which is computed at a clamped nuclear configuration, can be added to the BO PES resulting in a mass-dependent, so-called adiabatic potential. I have tried to follow another strategy which allows computation of electronic energies at fixed internuclear distances using the proper finite nuclear masses while maintaining the notion of the potential energy curve.

4 II. METODS APPLIED Since my main research interest was to develop techniques for solving the nuclear motion problem, I mostly employed my own computer codes written in FORTRAN. The newly developed DOPI program was applied in triatomic variational vibrational calculations. This code was also used for computing (ro)vibrationally averaged properties of triatomic molecules. Another FORTRAN code I developed, based on the use of Bessel-DVR basis functions, was employed for the determination of the (ro)vibrational energy levels if the singularities in the amiltonian became important. I used the symbolic algebra package Mathematica for deriving formulae and performing numerical integrations required for my non-bo calculations. The standard electronic structure program packages PSI and ACESII were employed for the determination of DBOCs and relativistic corrections, respectively. III. RESULTS AND CONCLUSIONS Variational calculation of vibrational energies and (ro)vibrationally averaged properties of triatomic molecules 1. A simple variational strategy termed DOPI was developed for solving the triatomic vibrational problem. DOPI stands for using direct product (P) DVR (D) basis for setting up the matrix representation of the amiltonian expressed in orthogonal (O) internal coordinates, e.g. Jacobi or Radau coordinates, and the requested eigenvalues and eigenvectors of the sparse amiltonian matrix of special structure are computed by an iterative (I) eigensolver. The DOPI code has been employed for computing vibrational energy levels for several molecules, such as 2 O, CO 2, N 2 O, C 2, CCl 2, CCl,, and CN/NC. 2. Since (semi)global PESs are available only for small molecules, the PESs of larger systems are usually given as a truncated Taylor series. Therefore, it is worth studying the utility of different anharmonic force field representations. Quartic and sextic force fields were used in variational calculations for computing vibrational energy levels of 2 O, CO 2, and N 2 O. The main conclusion is that the PES can be well represented using a quartic force field if the SPF coordinates are employed for describing the stretching motion. Furthermore, in the case of molecules containing hydrogen atom(s), e.g., CCl 4

5 in my study, it is worth augmenting the quartic force field with quintic and sextic diagonal bending force constant in order to improve the description of the bending motion.. Equilibrium structures of the water molecule were determinated. The best massindependent so-called BO equilibrium bond length and bond angle of 2 O is Å and , respectively. The isotope-dependent, so-called adiabatic equilibrium bond lengths are Å and Å and the bond angles are and for 16 2 O and D 16 2 O, respectively. 4. The temperature-dependent effective r g structure of the water molecule was also computed variationally. The r g parameters are the thermally averaged values of the internuclear distances. The r g r e values of the O- distances at 00 K are Å and Å for 16 2 O and D 16 2 O, respectively. This is a nice demonstration of the fact that a huge isotope effect can be observed in the vibrationally averaged parameters. The variational computation of the averaged properties allows dealing exactly with the rotational contribution to the effective parameters. The above-mentioned r g (O) r e (O) value for 16 2 O is Å if the rotational contribution is not neglected. It was also shown that the rotational contribution has a linear temperature dependence. Vibrationally averaged rotational constants were also computed variationally. It was shown that not the principal axes system (PAS) but the Eckart system should be employed for these calculations. Treating singularities in triatomic variational (ro)vibrational calculations 5. A method was developed for treating singularities in triatomic rotational-vibrational calculations using a nondirect product basis. This basis is formed employing the Bessel- DVR functions developed by Littlejohn and Cargo. Since the Bessel-DVR functions are not polynomials, the so-called generalized finite basis representation (GFBR) was employed for computing the matrix representation of the amiltonian. This study is a nice demonstration of the utility of the GFBR methods. 6. An efficient FBR was developed which can be employed with nondirect product bases having structure similar to that of spherical harmonics. Since the nondirect product basis employed for treating the singularities present in the (ro)vibrational kinetic energy operators given in orthogonal coordinate systems has a structure similar to that of spherical harmonics, the newly developed efficient FBR could be employed for 5

6 improving the efficiency of the (ro)vibrational calculations. The new method was applied for computing the (ro)vibrational energy levels of the molecular ion above its barrier to linearity. 7. An algorithm was introduced for treating all the singularities using a direct product basis having the proper boundary conditions. Exact matrix elements of the singular radial operator were given using a Bessel-DVR basis. The efficiency of the proposed method was improved by employing potential optimized Bessel-DVR. Vibrational energy levels of near the dissociation limit are presented. Beyond the Born Oppenheimer approximation 8. A variational method was developed for solving the Schrödinger equation of 2 -like systems using Jacobi coordinates and treating the distance of the two nuclei as a parameter. This procedure allows performing electronic energy calculations at fixed internuclear separations using the proper finite nuclear masses. Thus, a new adiabatic correction, the so-called adiabatic Jacobi correction (AJC) is defined as a difference between energies obtained from calculations with finite and infinite nuclear masses. AJCs were computed for the molecular ions 2, D 2, and D at different internuclear separations. DBOCs were also calculated and were found to be larger than the AJCs. Expectation values of the nucleus-electron distances were also computed and compared to the results obtained from fully nonadiabatic calculations. Unlike the BO treatment, this new adiabatic method is able to follow the symmetry breaking in D, similarly to nonadiabatic methods. Computations were performed for counterfactual systems changing the electron rest mass and the elementary charge as well as the mass and charge of the electron. It was shown that a decrease in the proportion of the nuclear and electronic mass would mean the breakdown of the BO approximation. Furthermore, an increase in both the mass of the proton and that of the electron without changing the proportion of the two masses would also result in the failure of the BO approximation. Finally, all the vibrational energy levels of curves at different levels of sophistication. 2 have been given using potential energy 6

7 IV. PUBLICATIONS Research articles [1] V. Szalay, G. Czakó, Á. Nagy, T. Furtenbacher, and A. G. Császár On one-dimensional discrete variable representations with general basis functions, J. Chem. Phys. 119, (200) [2] G. Czakó, T. Furtenbacher, A. G. Császár, and V. Szalay Variational vibrational calculations using high-order anharmonic force fields, Mol. Phys. 102, 2411 (2004) [] G. Czakó, V. Szalay, A. G. Császár, and T. Furtenbacher Treating singularities present in the Sutcliffe-Tennyson vibrational amiltonian in orthogonal internal coordinates, J. Chem. Phys. 122, (2005) [4] A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S. V. Shirin, N. F. Zobov, and O. L. Polyansky On equilibrium structures of the water molecule, J. Chem. Phys. 122, (2005) [5] Gy. Tarczay, T. A. Miller, G. Czakó, and A. G. Császár Accurate ab initio determination of spectroscopic and thermochemical properties of mono- and dichlorocarbenes, Phys. Chem. Chem. Phys. 7, 2881 (2005) [6] T. Furtenbacher, G. Czakó, B. T. Sutcliffe, A. G. Császár, and V. Szalay The methylene saga continues: stretching fundamentals and zero-point energy of X B 1 C 2, J. Mol. Struct , 28 (2006) [7] G. Czakó, V. Szalay, and A. G. Császár Finite basis representations with nondirect-product basis functions having structure similar to that of spherical harmonics, J. Chem. Phys. 124, (2006) [8] A. G. Császár, T. Furtenbacher, and G. Czakó The greenhouse effect on Earth and the complete spectroscopy of water, Magy. Kém. Foly. 112, 12 (2006) [9] G. Czakó, A. G. Császár, V. Szalay, and B. T. Sutcliffe Adiabatic Jacobi corrections for 2 -like systems, J. Chem. Phys. 126, (2007) [10] A. G. Császár, G. Czakó, T. Furtenbacher, and E. Mátyus An active database approach to complete rotational-vibrational spectra of small 7

8 molecules, Ann. Rep. Comp. Chem. in print (2007) [11] G. Czakó, T. Furtenbacher, P. Barletta, A. G. Császár, V. Szalay, and B. T. Sutcliffe Use of a nondirect-product basis for treating singularities in triatomic rotationalvibrational calculations, Phys. Chem. Chem. Phys. in print (2007) [12] E. Mátyus, G. Czakó, B. T. Sutcliffe, and A. G. Császár Vibrational energy levels with arbitrary potentials using the Eckart Watson amiltonian and the discrete variable representation, J. Chem. Phys. submitted (2007) [1] W. D. Allen, L. Woodcock. F. Schaefer III, I. N. Kozin, E. Mátyus, G. Czakó, and A. G. Császár Variational computation of the rovibrational states of isocyanic acid, in preparation (2007) [14] G. Czakó, T. Furtenbacher, A. G. Császár, and V. Szalay Temperature-dependent effective structures of the water molecule, in preparation (2007) Conference proceedings [1] A. G. Császár, V. Szalay, and G. Czakó First-principles rovibrational spectroscopy, The Nineteenth International Course & Conference on the Interfaces among Mathematics, Chemistry & Computer Sciences, Dubrovnik, Croatia, June 2004 [2] G. Czakó, A. G. Császár, and V. Szalay Toward first-principles complete spectroscopy of small molecules, The 2005 Younger European Chemists Conference, Brno, Czech Republic, September 2005 [] G. Czakó, V. Szalay, and A. G. Császár Szingularitások kezelése háromtest problémák variációs megoldásai során, XI. Anyagszerkezet-kutatási Konferencia, Mátrafüred, ungary, May 2006 [4] T. Furtenbacher, G. Czakó, A. G. Császár, and V. Szalay Kismolekulák teljes spektroszkópiája, XI. Anyagszerkezet-kutatási Konferencia, Mátrafüred, ungary, May 2006 [5] E. Mátyus, G. Czakó, B. T. Sutcliffe, and A. G. Császár Variational vibrational calculations in normal coordinates with arbitrary potentials, Molecular Quantum Mechanics: Analytic Gradients and Beyond, An International Conference in onor of Professor Péter Pulay, Budapest, ungary, May

9 [6] G. Czakó, T. Furtenbacher, A. G. Császár, and V. Szalay Toward first-principles complete spectroscopy of small molecules, Molecular Quantum Mechanics: Analytic Gradients and Beyond, An International Conference in onor of Professor Péter Pulay, Budapest, ungary, May 2007 [7] G. Czakó, Cs. Fábri, A. G. Császár, V. Szalay, B. T. Sutcliffe, and Gy. Tasi Adiabatic Jacobi corrections for 2 -like systems, Molecular Quantum Mechanics: Analytic Gradients and Beyond, An International Conference in onor of Professor Péter Pulay, Budapest, ungary, May