Variational calculations of the spectra of anharmonic oscillators using displaced gaussian functions

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1 Variational calculations of the spectra of anharmonic oscillators using displaced gaussian functions Jozef Motyčka 1 and Peter Babinec Department of Nuclear Physics and Biophysics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina F1, 84 48, Bratislava, Slovak Republic 1 jozef.motycka@gmail.com, babinec@fmph.uniba.sk Abstract. A straightforward variational method is presented for the accurate and simple calculations of the upper bounds to ground and excited energy states of one dimensional quantum anharmonic oscillators. The method is easily implemented in Matlab and its efficiency illustrated on several well known examples, like quartic, sextic and octic oscillators. 1. Introduction Oscillator considered as one of the simplest quantum mechanical system is described by Hamiltonian [1], [] : Ĥ = m + 1 m ω x (1) The harmonic potential standing on left-hand side allows to solve this eigenvalue problem in analytical way. General form of function belonging to spectrum of Hamiltonian (1) which are non-degenerate is written as follows ( m ω0 ) Ψ n (x) = π n n! H n (ξ) exp ξ where ξ = x, x 0 = x 0 m ω and the corresponding eigenvalues for ground and excited states are [1] ( E n = ω n + 1 ) for n = 0, 1,... (3) In the case when we update potential to anharmonic one (standard maner is to put following term λ x 4 ), there is no way to find analytical solution so we need turn to approximate methods. One of the basic methods - variational approximation emerges in ()

2 Variational calculations of the spectra of anharmonic oscillators... other well known cornerstone approximations used in quantum chemistry (e.g. Hartree- Fock) and offers advantage to dealing with this tasks. Results of variational approach depends mainly on choosing appropriate trial function. Generally variational method is based upon principle, that with given normalized wave function (futher bellow called trial function) which can be defined as linear combination of eigenfunctions of ((1)) the mean value of Hamiltonian for this state represents a upper bound to energy of ground state [3] [4]. That is, using bracket notation [5] Ψ trial Ĥ Ψ trial E 0 (4) Furthermore, we have to employ function orthogonal to ground state as a trial one to obtain upper bounds to energy of first excited state Ψ trial Ĥ Ψ trial E 1 (5) The normalized trial function depending on special parameters undergoes variations with respect to them in order to reach minimum of mean value of Ĥ as a good estimated ground energy of system. The trial function can be approximated as linear combination of set of functions, whereas we don t have to suppose that they are orthogonal or even normalized [4] N Ψ trial = c n ψ n (6) n=1 where c n are now variational parameters. Rayleigh-Ritz method is very useful variant to general variational method, which suggests more flexible trial function as given (6). The deal is to minimize the energy as a functional of trial function E [Ψ trial ] = Ψ trial Ĥ Ψ trial = ij ψ i Ĥ ψ j (7) and simultaneously with normalization condition to be fullfilled Ψ trial Ψ trial 1 = ij c i c j ψ i ψ j 1 = 0 (8) We define new function L using Lagrange multiplier ϵ L = ( ) c i c j ψ i Ĥ ψ j ϵ c i c j ψ i ψ j 1 ij ij and instead of solving δe = 0 (i.e. find that optimum trial function for which this first order variation vanish) one looks for (9) δl = δc i c j ψ i Ĥ ψ j (10) ij ( ) ϵ δc i c j ψ i ψ j 1 ij + complex conjugate = 0

3 Variational calculations of the spectra of anharmonic oscillators... 3 and with introducing notation ( ψ 1 ψ j = S ij - overlap inegral and ψ 1 Ĥ ψ j = H ij - matrix element) and by recognizing the arbitreness of δc i we finally get to expression also rewritten in matrix fashion H ij c j = ϵ S ij c j (11) j j H c = ϵ S c (1) The Lagrange multipliers ϵ rise here as sought-after bounds to energy. Non-trivial solutions (c column vector) for this set of linear equations are obtained for ϵ subjected to condition det (H ϵ S) = 0 (13). Computational details In present paper we also deal with the simplest form of harmonic oscillator Hamiltonian with exact spectrum E n = n + 1/ for n = 0, 1,... Ĥ = x (14) values of all constants are initially set to 1. We take into account effect of the quartic λ x 4, sextic λ x 6 and octic λ x 8 term included separatelly as well as together into hamiltonian (14). Ĥ = x + λ x 4 (15) Ĥ = x + λ x 6 (16) Ĥ = x + λ x 8 (17) and use same two types of trial function mentioned above. Pertubation constant λ is set to different values. Mainly quartic oscillator has been extensivelly studied sincethe early 1970s due to its equivalence to the one-dimensional φ 4 quantum field theory, because detailed study of this simplified system should shed tight on the structure of the φ 4 theory in higher dimensions. Pure quartic sextic and octic potentials are also considered Ĥ = 1 + λ x4 (18) Ĥ = 1 + λ x6 (19) Ĥ = 1 + λ x8 (0)

4 Variational calculations of the spectra of anharmonic oscillators... 4 Eigenvalues obtained from variational approach were compared with those resulting from diagonalisation of matrix representation of corresponding Hamiltonians. We begin with trial function, which is linear superposition of Gaussian type trial functions symetrically displaced from the origin (see figure 1 (top)) Ψ trial = i { c i1 exp [ (x i x 0) ] + c i exp [ (x + i x 0) ]} and tackle a problem of finding the best groung state energy estimation and also optimal parameters c 1 and c using Rayleigh-Ritz method. Number of pairs of gaussian functions varies from 1 to 4. The centre of Gaussian functions x 0 is chosen x 0 = 1, 1, We also try slightly different form of trial function (see figure (top)) Ψ trial = [ ] [ ] } c i {exp (x i x 0) + exp (x + i x 0) () i and number of included pairs of gaussian functions i, j vary from 1 to 4. Futher we employ trial functions Ψ trial and Ψ trial relative to (1) and () orthogonal to anticipated form of exact ground state function, i.e. to C exp x using our specific units (see(14) ħ and it is clear that x 0 = = 1), where C is for now normalization constant. So mω we try the following ansatz (1) and Ψ trial = { [ ] c i1 (x i x 0 ) exp (x i x 0) i + c i (x + i x 0 ) exp [ (x + i x 0) ]} Ψ trial = [ ] c i {(x i x 0 ) exp (x i x 0) i [ ] } + (x + i x 0 ) exp (x + i x 0) (3) (4) 3. Results and Discussion Computed energies and variational coefficients are sumarized in tables in section Tables and table captions, when particular trial functions (1) and () are used and λ = 1. table 1 and table on page 10 provide information about harmonic oscillator while table 4 and table 3 about anharmonic variant. The upper bounds to ground state energies approaches typical value (0.5 in our setup units) (as i is rising) when harmonic potential is considered. In a case of trial function () there is a rapid convergence as compared with function (1).

5 Variational calculations of the spectra of anharmonic oscillators... 5 Influence of changing pertubation constant on ground state eigenvalues for quartic and pure quartic anharmonic oscillators with Hamiltonian forms (15) and (18) is reported in table 5 and table 6. In point of non-pure potential table 5, there is a significant increase in relative error when constant λ > 1. Variational functions of type (1) and () with i = 3 and i = 6 used in respectivelly, because we d like to draw comparison two variants of trial functions by calculating the ground eigenvalues from equally sized determinant, i.e. for example in our case determinant calculated from 6 6 matrix. Exact eigenvalues are reference values obtained by diagonalization of relevant enough large matrix of size and considered as the lowest ones for different λ. For pure quartic potential table 6 relative error tend to rise extremaly when pertubation constant run over value 1. But in both cases, pure and non-pure oscillators we can see that error is significantly smaller (for any λ for trial functions () than for (1). Finally we employ also sextic, pure sectic, octic and pure octic potentials. The results are sumarized in table 7, table 8, table 9, table 10. Notes mentioned above for quartic and pure quartic potentials are valid for these cases, but the relative error gains higher values than in quartic and pure quartic and even reaches over 50 percent, when λ =. First excited state energies for various λ are sumarized in table 11. Constants ϵ 1 represents energy calculated using orthogonal trial functions (3) and (4) and ϵ energy obtained from variation of ground state estimation (1), () as second-lowest value. The relative error is taken for ϵ 1. Clearly, as regards trial functions (1) comparing to () there is always major error. Results previous mentioned for trial ground states, associated with increasing value of λ arise herein too. Nevertheless the later reveal its first excited state energies to differ distinctly from the exact. The variational functions for harmonic oscillator are shown up in following pictures. On first two pictures there are four trial functions for diferent number of expanding gaussian functions (1) (from i = 1 to i = 4) before and after variation procedure is applied. Same holds for second pair of pictures except we deal with ().

6 Variational calculations of the spectra of anharmonic oscillators i =1 i = i =3 i = x i =1 i = i =3 i = x Figure 1: Trial functions of form (1) for harmonic oscillator for i from i = 1 to i = 4 before variation (top) and after variation (bottom)

7 Variational calculations of the spectra of anharmonic oscillators i =1 i = x i =1 i = i =3 i = x Figure : Trial functions of form () for harmonic oscillator with i = 1 and i = 4 before (normalized) (top) and after variation there are all pairs for i from i = 1 to i = 4 of Gaussian functions included (bottom) Other following pictures give us insight into shapes of trial functions after variation, when anharmonic oscillator is studied (compare with figure 1 and figure )

8 Variational calculations of the spectra of anharmonic oscillators i =1 i = i =3 i = x i =1 i = i =3 i = x Figure 3: Trial functions of form (1) (top) and () (bottom) for anharmonic oscillator with i = 1 i = 4 and for λ = 1 after variation

9 Variational calculations of the spectra of anharmonic oscillators... 9 References [1] J. Pišút, L. Gomolčák, and V. Černý. Úvod do kvantovej mechaniky. Alfa, [] A. Böhm. Quantum mechanics. Springer, New York, [3] P. Atkins and R. Friedman. Molecular Quantum Mechanics. Oxford University Press Inc., New York, 4. edition, 005. [4] J. M. Ostlund and A. Szabo. Modern Quantum Chemistry. Dover Publications, Inc, [5] L. Piela. Ideas of Quantum Chemistry. Elsevier, 1. edition, 007.

10 Variational calculations of the spectra of anharmonic oscillators Tables and table captions Table 1. Energies and related coefficients for case of harmonic oscillator and trial functions (1) i ϵ coefficients c 11 = c 1 = c 11 = c 1 = c 1 = c = c 11 = c 1 = c 1 = c = c 31 = c 3 = c 11 = c 1 = c 1 = c = c 31 = c 3 = c 41 = c 11 = Table. Energies and related coefficients for case of harmonic oscillator and trial functions () i ϵ coefficients c 1 = c = c 1 = c = c 3 = c 1 = c = c 3 = c 4 =

11 Variational calculations of the spectra of anharmonic oscillators Table 3. Energies and related coefficients for case of quartic anharmonic oscillator (15) for λ = 1 and trial functions (1) i ϵ coefficients c 11 = c 1 = c 11 = c 1 = c 1 = c = c 11 = c 1 = c 1 = c = c 31 = c 3 = c 11 = c 1 = c 1 = c = c 31 = c 3 = c 41 = c 4 = Table 4. Energies and related coefficients for case of quartic anharmonic oscillator table 15 for λ = 1 and trial functions () i ϵ coefficients c 1 = c = c 1 = c = c 3 = c 1 = c = c 3 = c 4 =

12 Variational calculations of the spectra of anharmonic oscillators... 1 Table 5. Energies for quartic anharmonic oscillator (15) for different values of λ and trial functions of types (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) () (1) () (1) () Table 6. Energies of pure quartic anharmonic oscillator table 18 using trial functions (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) () (1) ()

13 Variational calculations of the spectra of anharmonic oscillators Table 7. Energies for sextic anharmonic oscillator (15) for different values of λ and trial functions of types (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) () Table 8. Energies for pure sextic anharmonic oscillator (15) for different values of λ and trial functions of types (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) ()

14 Variational calculations of the spectra of anharmonic oscillators Table 9. Energies for octic anharmonic oscillator (15) for different values of λ and trial functions of types (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) () Table 10. Energies for pure octic anharmonic oscillator (15) for different values of λ and trial functions of types (1) for i = 3 and () for i = 6 λ trial function ϵ (variational) ϵ (exact) relative error (%) (1) () (1) () (1) () (1) ()

15 Variational calculations of the spectra of anharmonic oscillators Table 11. Energies for excited states of harmonic and quartic anharmonic oscillator for different λ using trial functions (3) (4); ϵ 1 is lowest energy for first excited state estimations, ϵ is second-lowest energy for ground state estimations (1) for i = 3 ans () for i = 6; relative error is calculated for ϵ 1 λ trial function ϵ 1 ϵ ϵ (exact) relative error (%) (3) (4) *e-5 (3) (4) (3) (4) (3) (4)