Quantum Final Project by Anustup Poddar and Cody Tripp 12/10/2013

Transcription

1 Quantum Final Project by Anustup Poddar and Cody Tripp Introduction The Hamiltonian in the Schrӧdinger equation is the sum of a kinetic and potential energy operator. The Fourier grid Hamiltonian (FGH) method relies on the momentum representation of the kinetic energy operator and the coordinate representation for the potential energy operator. Kosloff 1 has shown that these ideas can be used to evaluate the expression ĤY. Kosloff represented both Y and ĤY as vectors whose components are the values of the function on a grid of points in coordinate space. In the paper we chose, Marston and Balint-Kurti 2 identify the matrix representation of Ĥ in this vector space. The calculation of the Ĥ matrix in this space is relatively simple. To calculate Ĥ, the potential is evaluated at the grid points and then a forward and reverse Fourier transform reduces the expression to a summation over cosine functions. This will be explained in the theory section of the paper. The FGH method is a special case of a discrete variable representation (DVR) originally introduced by Harris et al. in There have been numerous DVR techniques since Harris, but the FGH method has the advantage of simplicity over the other techniques. The simplicity of the process lies in the fact that the eigenfunctions of the Hamiltonian operator are generated directly as amplitudes of the wavefunctions on the grid points. This is in contrast to many other techniques that give the eigenfunctions as linear combinations of basis set functions. There are similarities to the variational methods we have done in class and the FGH method, which will become evident in the theory sections below. Theory The non-relativistic Hamiltonian is: th= t tp 2 T+V(tx)= +V(tx) 2m (1) We want to manipulate this Hamiltonian into a form that can be used in our matrix representation, but first some definitions are needed. Eqn. 2 is a standard orthogonality relationship in quantum mechanics which reduces to the Kronecker delta. Eqn. 2 is in terms of coordinates. < x x l>=d(x- x) l An identity relationship for momentum is also defined in eqn. 3. # ti k= k>< k dk (3) The transformation matrix elements between the coordinate and momentum representations can be derived as follows: (2)

2 tp=- i' 2 2x tpw k(x)='kw k(x) &- i' 2 2x W k(x)='kw k(x) &2W k(x)=ikw k(x)2x 2W k(x) & = ik2x W k(x) &W k(x)=e ikx (4) ow Ψ k (x) is the projection of the momentum vector, k, on the coordinate vector, x, represented as: W k(x)=< x k> The normalization constant,, can be calculated from the relation: # e ikx dx=d(k) Which gives =. Therefore, the forward and the reverse Fourier Transform can then be represented as: < x k>= 1 e ikx < k x>= 1 e -ikx The potential is the diagonal in the coordinate representation as shown in eqn x l V(tx) x2=v(x)d(x- x) l ow we can find out the expectation value of the Hamiltonian from eqn. 1 as: < x H t x l>=< x ( T+V(tx)) t x l> =< x T t x l>+<x V(tx) x l> &<x th l x >=< x t T l x >+V(x)d(x- l x) The identity relationship from eqn. 3 can now be inserted into eqn. 10. < x t H l x >=< x t T k>< k Since we have the eigenvalue equation as: G # J l x > dk+v(x)d(x- l x) (5) (6) (7 & 8) (9) (10) (11)

3 tt k>= T k k> eqn. 11 reduces to: < x th x l>= # < x k> T k< k x l> dk+v(x)d(x- x) l (12) (13) We now use a forward and reverse Fourier transform (eqns. 7 & 8) to further simplify the equation to exponential from: Discretization < x H t x l>= 1 e ik(x- x) l # T kdk+v(x)d(x- l The Fourier grid Hamiltonian method transforms the continuous range of coordinate values to a grid of discrete values, x i, where, x i =iδx. Here Δx is the uniform spacing between the grid points in coordinate space and i is the index number of the grid points. We use grid values for x. The total length of the coordinate space is Δx. As a result of this discretization, we have to re-define of the normalization condition of the wave function: } * # (x)}(x)dx=1 } * (x i) }(x i)dx=1 Dx } i 2 =1 (14) becomes i=1, or, i=1 (15) since Ψ i = Ψ(x i ) is the value of the wave function at the grid point i. The total length of the coordinate space represents the maximum wavelength and therefore the smallest frequency in the reciprocal momentum space. Therefore, we may say that the spacing between the grid points in momentum space, Δk, can be represented as x) = = (16) ow, quantum particles can either move to the right or to the left, i.e., positive or negative k. So here we consider that the momentum space is centered at k=0, with n grid values in both the positive and the negative directions, where 2n=-1. Therefore, we have a relation similar to our coordinate space, k=lδk, where l is the index number of the grid points. In the paper by Marston and Balint-Kurti the FGH method must employ an odd number of grid points,. In our new coordinate space we can use bra-ket notation to show the values of the wave function at specific grid points: < x i }>=}(x i)=}i (17)

4 Then the next task is to redefine orthogonality and identity relations for our new coordinate space: i=1 ti x= x i>dx<x i Dx<x i x j>=d (19) Armed with the needed machinery we now proceed to find the discretized analog of our Hamiltonian in eqn. 14. < x th x l>= 1 e ik(x-x) l # T kdk+v(x)d(x- l x) We use the relations k=lδk, eqn. 19, and the relation for kinetic energy, T k : (18) (20) Tl= & 2 k 2 = & 2 2m 2m (ltk) (21) We represent the kinetic energy as T l because we now use the l-index in our equation. We then convert the integration from negative infinity to positive infinity into a sum from n to +n: H =< x i H t x j> n = 1 e il Dk(x i - x j ) ' 2 G 2m (ldk) 2 JDk+ l=-n V(x i)d Dx We then use the relation between Δk and Δx (eqn. 16), to further simplify our equation to: (22) H = 1 or,h = 1 Dx S G n Tl l=-n +n e il2 r(i-j) Tl l=-n Dx X e [il(2 r Dx)(i-j)Dx] F I+ F I+ V(x i)d J V(x i)d Dx (23) We now use Euler s formula, = +, for both the positive and negative values of l. The sine terms being both positive and negative cancel out and with T 0 being zero (eqn. 21), we are left with a sum of cosines. H = 1 Dx or,h = 1 Dx n G 2cos(l(i-j)) T l+ V(x i)d J l=1 n G2 2cos(l(i-j))Tl+ V(x i)d J l=1 (24)

5 Using the above equation we can now find out the eigenvalues and amplitudes of the wavefunctions at the grid points utilizing the variational method. We use the identity relation (eqn. 18) and the fact that < > = (which is the amplitude of the wave function at the particular grid point i) to find a discrete analog of our wave function. i=1 }>=t I x }>= x i>dx<x i }> = x i>dx} i i=1 (25) The expectation value of the energy for the arbitrary wave function can now be written using the orthogonality relation (eqn. 19): } * i Dx<x i H x j>dx} j <} H }> E= = <} }> } * i Dx<x i x j>dx} j } * i DxH Dx} j = Dx } * i d } j (26) Then, a renormalized Hamiltonian matrix is defined for the wave function: H 0 = G2 n 2cos(l(i-j))Tl+ V(x i)d J= H Dx l=1 (27) Therefore, the expression of the expectation value of the energy changes: } * i DxH Dx} j E= Dx } * i d } j = } * i DxH } j } i * } jd = } * i H 0 } j } i * } jd (28) According to the variational method we minimize this energy, taking the derivatives with respect to the Ψ j (which here acts as our coefficients, unlike the actual coefficients of the wave functions in the normal variational principle) and as a result we get a set of secular equations: " H 0 - Emd % } m j=0 j (29) In the above equation, λ represents the quantum number of the energy level for the particular

6 system. The energies calculated are below the dissociation energy, V(x), of the system and that gives us the bound state energies. The eigenvector, Ψ l λ, gives us the amplitudes of the wave function at particular grid points. umerical Implementation The potential of the Morse curve is modeled by an equation of the form V= D[1- e ( - b (x-x e) ] 2 (30) where D has units of energy and is a scaling factor that dictates the horizontal asymptote the potential approaches as x. If VD is plotted, the Morse potential approaches 1 as x. This is the form of Figures 1-3 below. β is in units of inverse distance and sets the width of the Morse potential. The variable x e is in units of distance and is the equilibrium bond distance for the molecule of interest. These three parameters will change the shape of the Morse potential depending on the molecule of interest. The parameters for H 2 are given in Table 1. Table 1. Morse parameters for H 2 in atomic units (a.u.) D = a.u. = ev β = a.u. = x10 10 m -1 x e = a.u. = x10-10 m The Morse parameters, along with the grid size and maximum x-value, are the only inputs needed to get approximate values for the energies and wavefunction amplitudes. We have chosen a grid size of =129, which is what Marston uses in the paper. Only energies below the Morse dissociation energy are valid, so an estimate of the number of valid energy levels is needed: ymax= 2Dnb=17.42 (31) The maximum vibrational quantum number was taken to be 16 (the 17 th state). The maximum x-value was taken to be 1.5 times larger than the outer turning point of the 16 th eigenstate which ensures that all of the relevant features are captured. This value comes out to x The maximum value of x is then split into uniform steps: x i=i3x, 0# i# -1 (32) The energy levels for vibrational quantum numbers 0-16 and the deviation from the analytical values is shown in Table 2. When taken to 8 digits, the energy values calculated by Marston deviate from the analytical values by an average of 0.3 ppm. Our calculations had higher deviations. The smallest deviation was 9 ppm, while the largest was 102 ppm. The average of the deviations was 65 ppm. The deviation becomes progressively smaller as ν 16.

7 Although the calculated deviations were a factor of 200 larger than Marston s, the overall accuracy is still high, on the order of ppm. Table 2. Eigenvalue energies (in Hartree) calculated with a 129x129 grid comparing results with both Marston and the analytical Morse energies Poddar and Tripp Marston Vibrational Quantum umber Energy (a.u.) Deviation (ppm) Energy (a.u) Deviation (ppm) Exact Avg. Deviation (ppm) References (1) Kosloff, R.; Tal-Exer, H. A Direct Relaxation Method for Calculating Eigenfunctions and Eigenvalues of the Schrodinger Equation on a Grid. Chem. Phys. Lett. 1986, 127, (2) Marston, C. C.; Balint-Kurti, G. G. The Fourier Grid Hamiltonian Method for Bound State Eigenvalues and Eigenfunctions. J. Chem. Phys. 1989, 91, 3571.

8 (a) (b) Figure 1. H 2 wavefunctions for ν=0 with Morse potential and energy level: (a) from Marston 2 ;(b) reproduced using Marston s parameters. (a) (b) Figure 2. H 2 wavefunctions for ν=5 with Morse potential and energy level: (a) from Marston 2 ; (b) reproduced using Marston s parameters. (a) (b) Figure 3. H 2 wavefunctions for ν=15 with Morse potential and energy level: (a) from Marston 2 ; (b) reproduced using Marston s parameters

9 Supplemental Figures Deviation from Exact Energy Values for 129x129 Grid Error (ppm) Vibrational Quantum umber 600 Processing Time for Grid Size Time (s) Grid Length

10 Supplemental Figures 60 Average Percent Error in Calculated Energies 50 Average Percent Error Grid Length

11 Division of Labor Both Anustup and I worked closely on all aspects of the project.