Introduction to numerical projects

Introduction to numerical projects Here follows a brief recipe and recommendation on how to write a report for each project. Give a short description of the nature of the problem and the eventual numerical methods you have used. Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself. Include the source code of your program. Comment your program properly. If possible, try to find analytic solutions, or known limits in order to test your program when developing the code. Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes. Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc. Try to give an interpretation of you results in your answers to the problems. Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you ve made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it. We do appreciate your comments. Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don t properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning. Format for electronic delivery of report and programs The preferred format for the report is a PDF file. As programming language we prefer that you choose between C/C++ and Fortran. You could also use Java or Python as programming languages. Matlab/Maple/Mathematica/IDL are not accepted, but you can use them to check your results where possible. Finally, we do prefer that you work together. Optimal working groups consist of 2-3 students, but more people can collaborate. You can then hand in a common report. Below you will also find a guide on how to write up the project as a scientific article. We recommend this option when writing the final report. 1

Project for FYS4411/9411 Spring 2014, Advanced Hartree- Fock Part 1: Hartree-Fock calculations of closed-shell atoms The Hartree-Fock functional is written as E[Φ] = + µ=1 µ h µ 1 2 [ µ=1 ν=1 µν 1 r ij µν µν 1 r ij νµ With the given functional, we can perform at least two types of variational strategies. Vary the Slater determinant by changing the spatial part of the single-particle wave functions themselves. Expand the single-particle functions in a known basis and vary the coefficients, that is, the new function single-particle wave function a is written as a linear expansion in terms of a fixed basis φ (harmonic oscillator, Laguerre polynomials etc) ]. ψ a = λ C aλ φ λ, Both cases lead to a new Slater determinant which is related to the previous via a unitary transformation. The second one is the one we will use in this project. 1a) Consider a Slater determinant built up of single-particle orbitals ψ λ, with λ = 1, 2,..., N. The unitary transformation ψ a = C aλ φ λ, λ brings us into the new basis. Show that the new basis is orthonormal. Show that the new Slater determinant constructed from the new single-particle wave functions can be written as the determinant based on the previous basis and the determinant of the matrix C. Show that the old and the new Slater determinants are equal up to a complex constant with absolute value unity. (Hint, C is a unitary matrix). 1b) Minimizing with respect to Ckα, remembering that Ckα and C kα are independent and defining h HF αγ = α h γ + CaβC aδ αβ V γδ AS, a=1 show that you can write the Hartree-Fock equations as h HF αγ C kγ = ɛ k C kα. Explain the meaning of the different terms. γ 2 βδ

1c) We will now set up the Hartree-Fock program by using hydrogen-like single-particle functions for the 1s, 2s and 3s orbitals. We will apply these basis functions only to the helium and beryllium atoms. For all other systems, we will use so-called Gaussian type of orbitals (GTO). These will be introduced after we have set up our first Hartree-Fock programs. For Hydrogen like-orbitals, the matrix elements involving the Coulomb interaction, can all be evaluated in a closed form, see the table below for all integrals involving these orbitals. Your first step is to write a function which reads in these integrals and sets up the antisymmetrized matrix elements. (Hint: the table lists only the integrals, there is no spin). 1d) With the above ingredients we are now ready to solve the Hartree-Fock equations for the helium and beryllium atoms. Set up the Hartree-Fock equations for the ground states of helium and beryllium with the electrons occupying the respective hydrogen-like orbitals 1s, 2s and 3s. There is no spin-orbit part in the two-body Hamiltonian. Find the single-particle energies iteratively as explained in the slides and lecture notes. Find also the total binding ground state energy and compare this to the energy obtaining without performing the Hartree-Fock calculations. The experimental values are 2.904 atomic units (a.u.) and 14.67 a.u. for the helium and beryllium atoms, respectively. In setting up the equations make sure your Hartree-Fock matrix is block diagonal in the spin quantum numbers. Since there is no spin-dependent part in the Hamiltonian, states with the same l and n will be degenerate in energy with respect to the spin projections. Part 2: Hartree-Fock calculations of atoms and molecules using GTOs 2a) Our next steps involves an object orientation of the code and the inclusion of Gaussiantype of orbitals (GTOs) as detailed in the slides. Your task here is to repeat the calculations of the helium and beryllium atoms using GTOs using three and five primitive functions, a so-called STO-3G and STO-5L basis. Compare your results with those obtained with the hydrogen-like basis discussed above. Comment your results. 2b) Use thereafter a chosen GTO basis and compute the ground state energy of the neon and argon atoms and compare with existing theoretical calculations in the literature. Comment your results. 2c) We switch now to the computation of molecules and will use the hydrogen and beryllium molecules as our first test case with GTOs. The H 2 molecule consists of two protons and two electrons with a ground state energy E = 31.949 ev or 1.175 3

a.u. and the equilibrium distance between the two hydrogen atoms of r 0 = 1.40 Bohr radii (recall that a Bohr radius is 0.05 10 9 m. The Be 2 molecule contains eight electrons and eight protons (in addition we have eight neutrons but their charge is zero). Useful benchmarks for these molecules are the articles of Moskowitz and Kalos, Int. Journal of Quantum Chemistry XX, 1107 (1981) for He and H 2 and Filippi, Singh and Umrigar, J. Chemical Physics 105, 123 (1996) for the Be 2 molecule. You may also find Jørgen Høgberget s Master of Science thesis at the University of Oslo, 2013 a useful reference. 2d) Our final aim is to compute the ground state energy and equilibrium configuration for the water molecule H 2 O and Si 2 O using GTOs. Your task here is to figure out which is the optimal basis and perform calculations of the ground state energy and compare with the literature. With the optimal orbitals we can in turn also compute the charge distribution of electrons, which is given by the the so-called density distribution. Multiplying with the electrical charge gives the charge distribution. The density distribution is defined as ρ(r) = ρ(r 1 ) = dr 2... dr N Ψ(r 1, r 2... r N ) 2, which in a Hartree-Fock based approach becomes ρ(r) = ψ i (r) 2, i=1 with ψ the single-particle wave functions (our best basis). A useful check that the numerics works properly is to integrate the density distributions since it has to give us the total number of electrons in the system, that is N = drρ(r). Compute the density distribution for the H 2, Be 2, H 2 O and Si 2 O molecules at their equilbrium positions (the value of R where the ground state energy is at its minimum). Make plots of the density distributions. Give an interpretation of your results and discuss the physics behind the results. How to write a scientific report What should it contain? A possible structure If you wish to write up your findings as a scientific article, here are some simple guidelines An introduction where you explain the rational for the physics case and what you have done. At the end of the introduction you should give a brief summary of the structure of the report 4

Theoretical models and technicalities. This is the methods section. Results and discussion Conclusions and perspectives Appendix with extra material Bibliography You don t need to answer all questions in a chronological order. When you write the introduction you could focus on the following aspects A central aim is to study the role of correlations due to the repulsion between the electrons. To do this we have singled out several systems from atomic and molecular physics. We test also the wave functions by computing onebody densities. The methods section could contain the following elements: Describe the methods (quantum mechanical and algorithms) You need to explain Hartree-Fock theory. The trial wave functions. Why do you choose the functions you do? You can also plug in some calculations to demonstrate your code, such as selected runs from exercises 1a-1d for helium and beryllium. The results section should include results which support your conclusions. You don t need to present all results, focus on those which you find the most relevant. Provide also benchmarks which you have used in order to validate your code. Present your discussions of results and compare with other calculations and theories and give critical assessments of what you have done. In the concluding section you could focus on State your main findings and interpretations Try as far as possible to present perspectives for future work Try to discuss the pros and cons of the method and possible improvements Additional material, code listings etc can be placed in various appendices. Finally, you should provide a proper list of relevant references. Give always references to material you base your work on, either scientific articles/reports or books. When citing articles, refer as: name(s) of author(s), journal, volume (boldfaced), page and year. For books, refer as: name(s) of author(s), title of book, publisher, place and year, eventual page numbers. 5

Appendix Table 1 contains the matrix elements for the radial integrals to be used for the direct part and the exchange part. Note again that these integrals do not include spin. You will need these integrals in solving the Hartree-Fock equations for the helium and beryllium atoms when using hydrogen-like single-particle states. 6

Table 1: Closed form expressions for the Coulomb matrix elements. The nomenclature is 1 = 1s, 2 = 2s and 3 = 3s, with no spin degrees of freedom. 11 V 11 = (5Z)/8 11 V 12 = (4096 2Z)/64827 11 V 13 = (1269 3Z)/50000 11 V 21 = (4096 2Z)/64827 11 V 22 = (16Z)/729 11 V 23 = (110592 6Z)/24137569 11 V 31 = (1269 3Z)/50000 11 V 32 = (110592 6Z)/24137569 11 V 33 = (189Z)/32768 12 V 11 = (4096 2Z)/64827 12 V 12 = (17Z)/81 12 V 13 = (1555918848 6Z)/75429903125 12 V 21 = (16Z)/729 12 V 22 = (512 2Z)/84375 12 V 23 = (2160 3Z)/823543 12 V 31 = (110592 6Z)/24137569 12 V 32 = (29943 3Z)/13176688 12 V 33 = (1216512 2Z)/815730721 13 V 11 = (1269 3Z)/50000 13 V 12 = (1555918848 6Z)/75429903125 13 V 13 = (815Z)/8192 13 V 21 = (110592 6Z)/24137569 13 V 22 = (2160 3Z)/823543 13 V 23 = (37826560 2Z)/22024729467 13 V 31 = (189Z)/32768 13 V 32 = (1216512 2Z)/815730721 13 V 33 = (617Z)/(314928 3) 21 V 11 = (4096 2Z)/64827 21 V 12 = (16Z)/729 21 V 13 = (110592 6Z)/24137569 21 V 21 = (17Z)/81 21 V 22 = (512 2Z)/84375 21 V 23 = (29943 3Z)/13176688 21 V 31 = (1555918848 6Z)/75429903125 21 V 32 = (2160 3Z)/823543 21 V 33 = (1216512 2Z)/815730721 22 V 11 = (16Z)/729 22 V 12 = (512 2Z)/84375 22 V 13 = (2160 3Z)/823543 22 V 21 = (512 2Z)/84375 22 V 22 = (77Z)/512 22 V 23 = (5870679552 6Z)/669871503125 22 V 31 = (2160 3Z)/823543 22 V 32 = (5870679552 6Z)/669871503125 22 V 33 = (73008Z)/9765625 23 V 11 = (110592 6Z)/24137569 23 V 12 = (2160 3Z)/823543 23 V 13 = (37826560 2Z)/22024729467 23 V 21 = (29943 3Z)/13176688 23 V 22 = (5870679552 6Z)/669871503125 23 V 23 = (32857Z)/390625 23 V 31 = (1216512 2Z)/815730721 23 V 32 = (73008Z)/9765625 23 V 33 = (6890942464 2/3Z)/1210689028125 31 V 11 = (1269 3Z)/50000 31 V 12 = (110592 6Z)/24137569 31 V 13 = (189Z)/32768 31 V 21 = (1555918848 6Z)/75429903125 31 V 22 = (2160 3Z)/823543 31 V 23 = (1216512 2Z)/815730721 31 V 31 = (815Z)/8192 31 V 32 = (37826560 2Z)/22024729467 31 V 33 = (617Z)/(314928 3) 32 V 11 = (110592 6Z)/24137569 32 V 12 = (29943 3Z)/13176688 32 V 13 = (1216512 2Z)/815730721 32 V 21 = (2160 3Z)/823543 32 V 22 = (5870679552 6Z)/669871503125 32 V 23 = (73008Z)/9765625 32 V 31 = (37826560 2Z)/22024729467 32 V 32 = (32857Z)/390625 32 V 33 = (6890942464 2/3Z)/1210689028125 33 V 11 = (189Z)/32768 33 V 12 = (1216512 2Z)/815730721 33 V 13 = (617Z)/(314928 3) 33 V 21 = (1216512 2Z)/815730721 33 V 22 = (73008Z)/9765625 33 V 23 = (6890942464 2/3Z)/1210689028125 33 V 31 = (617Z)/(314928 3) 33 V 7 32 = (6890942464 2/3Z)/1210689028125 33 V 33 = (17Z)/256