0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 896.3 896.3 740.7 351.8 611.1 351.8 611.1 351.8 351.8 611.1 675.9 546.3 675.9 546.3 Variational methods for the solution of either the Schrödinger equation or its perturbation expansion can be used to obtain approximate eigenvalues and eigenfunctions of this Hamiltonian. /Type/Font 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Font To implement such a method one needs to know the Hamiltonian $$H$$ whose energy levels are sought and one needs to construct a trial wavefunction in which some 'flexibility' exists (e.g., as in the linear variational method where the $$a_j$$ coefficients can be varied). >> endobj /FontDescriptor 32 0 R AND B. L. MOISEIWITSCH University College, London (Received 4 August 1950) The variational methods proposed by … JOURNAL of coTATR)NAL PHYSICS 33, 359-368 (1979) Application of the Finite-Element Method to the Hydrogen Atom in a Box in an Electric Field M. FRIEDMAN Physics Dept., N.R.CN., P.O. Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter-mined as a combination of the various quantum "dynamical" analogues of The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. Question: Exercise 7: Variational Principle And Hydrogen Atom A) Variational Rnethod: Show That Elor Or Hlor)/(dTlor) Yields An Upper Bound To The Exact Ground State Energy Eo For Any Trial Wave Function . 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /BaseFont/UQQNXY+CMTI12 /Type/Font 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 In atomic and molecular problems, one common application of the linear variation method is in the configuration interaction method (CI).4 Here, with H usually the clamped nuclei Hamiltonian, the k are Slater determinants or linear combinations of Slater determinants, made out of given spin orbitals (the spin orbitals often also involving nonlinear parameters-- see end of Section 7). /BaseFont/DWANIY+CMSY10 of Physics, IIT Kharagpur Guide:Prof. Kumar Rao, Asst. /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 EXAMPLES: First, let’s use the Variation Method on some exactly solvable problems to see how well it does in calculating E0. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 935.2 351.8 611.1] 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 >> µ2. 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 This problem could be solved by the variational method by obtaining the energy of as a function of the variational parameter , and then minimizing to find the optimum value . /LastChar 196 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] /Name/F5 /Name/F2 ψ = 0 outside the box. 36 0 obj 1062.5 826.4] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 /Subtype/Type1 To determine the wave functions of the hydrogen-like atom, we use a Coulomb potential to describe the attractive interaction between the single electron and the nucleus, and a spherical reference frame centred on the centre of gravity of the two-body system. and for a trial wave function u jf ƔsՓ\���}���u���;��v��X!&��.y�ۺ�Nf���H����M8/�&��� << The application of variational methods to atomic scattering problems I. /Type/Font >> 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 The Stark effect on the ground state of the hydrogen atom is taken as an example. 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 << /FirstChar 33 endobj 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 H = … The book contains nine concise chapters wherein the first two ones tackle the general concept of the variation method and its applications. 2.1 Hydrogen Atom In this case the wave function is of the general form (8) For the ground state of hydrogen atom, the potential energy will be and hence the value of Hamiltonian operator will be According to the variation method (2.1) the energy of hydrogen atom can be calculated as /Name/F10 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 761.6 272 489.6] The first example applies the linear version of the variation method to the particle in a box model, using a basis with explicit parity symmetry, Phik(t) = N (1-t2)tk, where t = 2x/L -1 and N is the normalization constant. /Type/Font 1. 33 0 obj 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 The elastic scattering of electrons by hydrogen atoms BY H. S. W. MASSEY F.R.S. 1 APPLICATION OF THE VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS Suvrat R Rao, Student,Dept. application of variation method to hydrogen atom for calculation of variational parameter & ground state energy iit gate csir ugc net english /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 /Type/Font EXAMPLES: First, let’s use the Variation Method on some exactly solvable problems to see how well it does in calculating E0. This problem could be solved by the variational method by obtaining the energy of as a function of the variational parameter , and then minimizing to find the optimum value . 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /Subtype/Type1 /Name/F4 The ground-state energy of the N -dimensional helium atom is pre-sented by applying the variational principle. <> �#)�\�����~�y% q���lW7�#f�F��2 �9��kʡ9��!|��0�ӧ_������� Q0G���G��TME�V�P!X������#�P����B2´e�pؗC0��3���s��-��џ ���S0S�J� ���n(^r�g��L�����شu� /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 << /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 >> The use of hydrogen-powered fuel cells for ship propulsion, by contrast, is still at an early design or trial phase – with applications in smaller passenger ships, ferries or recreational craft. Let us attempt to calculate its ground-state energy. >> 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 << /FontDescriptor 8 0 R Box 9001, Beer Sheva, Israel A. RABINOVITCH Physics Dept., Ben Gurion University, Beer Sheva, Israel AND R. THIEBERGER Physics Dept., NACN., P.O. 694.5 295.1] The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. /FirstChar 33 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 18 0 obj << Some chapters deal with other theorems such as the Generealized Brillouin and Hellmann-Feynman Theorems. << 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 9 0 obj endobj /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter- mined as a combination of the various quantum "dynamical" analogues of classical motions (translation, vibration, rotation) and a central-force inter- action (i.e, the Coulomb interaction between an electron and a nucleus). << 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 /LastChar 196 >> 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 /Name/F7 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 endobj %PDF-1.3 We have to take into account both the symmetry of the wave-function involving two electrons, and the electrostatic interaction between the electrons. endobj 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /LastChar 196 /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 /LastChar 196 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 Some chapters deal with other theorems such as the Generealized Brillouin and Hellmann-Feynman Theorems. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 endobj For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 /Subtype/Type1 However, for systems that have more than one electron, the Schrödinger equation cannot be analytically solved and requires approximation like the variational method to be used. 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 /BaseFont/OASTWY+CMEX10 x��WKo�F����[����q-���!��Ch���J�̇�ҿ���H�i'hQ�d9���7�7�PP� 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 The interaction (perturbation) energy due to a field of strength ε with the hydrogen atom electron is easily shown to be: $E = \frac{- \alpha \varepsilon ^2}{2}$ Given that the ground state energy of the hydrogen atom is ‐0.5, in the presence of the electric field we would expect the electronic energy of the perturbed hydrogen atom to be, Remember, the typical hydrogen atom Hamiltonian looks like Hhydrogen = - ℏ2 2 m ∇2-e2 4 πϵ0 1 r (3.13) The second term has e2 in the numerator, but there it is 2 e2, because the nucleon of a helium atom has charge +2e. endobj 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 /BaseFont/MEAOQS+CMMI12 /FontDescriptor 29 0 R /Type/Font �����q����7Y������O�Ou,~��G�/�Rj��n� /Name/F1 To get some idea of how well this works, Messiah applies the method to the ground state of the hydrogen atom. /BaseFont/JVDFUX+CMSY8 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 826.4 295.1 531.3] If R is the vector from proton 1 to proton 2, then R r1 r2. << >> Remember, the typical hydrogen atom Hamiltonian looks like Hhydrogen = - ℏ2 2 m ∇2-e2 4 πϵ0 1 r (3.13) The second term has e2 in the numerator, but there it is 2 e2, because the nucleon of a helium atom … /Subtype/Type1 791.7 777.8] 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 /Name/F3 Helium Atom, Many-Electron Atoms, Variational Principle, Approximate Methods, Spin 21st April 2011 I. The Schrödinger equation can be solved exactly for our model systems including Particle in a Box (PIB), Harmonic Oscillator (HO), Rigid Rotor (RR), and the Hydrogen Atom. 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 /BaseFont/HLQJFV+CMR12 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] >> Hydrogen atom One example of the variational method would be using the Gaussian function as a trial function for the hydrogen atom ground state. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 /FirstChar 33 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 Each of these two Hamiltonian is a hydrogen atom Hamiltonian, but the nucleon charge is now doubled. /FontDescriptor 14 0 R The variational procedure involves adjusting all free parameters (in this case a) to minimize E˜ where: E˜ =< ψ˜|H|ψ>˜ (2) As you can see E˜ is sort of an expectation value of the actual Hamiltonian using 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 << 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 !� ��x7f$@��ׁ5)��|I+�3�ƶ��#a��o@�?�XA'�j�+ȯ���L�gh���i��9Ó���pQn4����wO�H*��i۴�u��B��~�̓4��JL>�[�x�d�>M�Ψ�#�D(T�˰�ͥ@�q5/�p6�0=w����OP"��e�Cw8aJe�]�B�ݎ BY7f��iX0��n�� _����s���ʔZ�t�R'�x}Jא%Q�4��0��L'�ڇ��&RX�%�F/��&V�y)���6vIz���X���X�� Y8�ŒΉሢۛ' �>�b}�i��n��С ߔ��>q䚪. The next four trial functions use several methods to increase the amount of electron-electron interactions in … 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 /BaseFont/GMELEA+CMMI8 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 /BaseFont/VSFBZC+CMR8 A new application of variational Monte Carlo method is presented to study the helium atom under the compression effect of a spherical box with radius (rc). (1) Find the upper bound to the ground state energy of a particle in a box of length L. V = 0 inside the box & ∞ outside. The book contains nine concise chapters wherein the first two ones tackle the general concept of the variation method and its applications. 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 /Type/Font 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5 Find the value of the parameters that minimizes this function and this yields the variational estimate for the ground state energy. >> Applications to model proton and hydrogen atom transfer reactions are presented to illustrate the implementation of these methods and to elucidate the fundamental principles of electron–proton correlation in hydrogen tunneling systems. This problem could be solved by the variational method by obtaining the energy of as a function of the variational parameter , and then minimizing to find the optimum value . /BaseFont/MAYCLP+CMBX12 For the Variational method approximation, the calculations begin with an uncorrelated wavefunction in which both electrons are placed in a hydrogenic orbital with scale factor $$\alpha$$. /Name/F9 m�ۉ����Wb��ŵ�.� ��b]8�0�29cs(�s?�G�� WL���}�5w��P�����mh�D���`���)~��y5B�*G��b�ڎ��! endobj The ground-state energies of the helium atom were calculated for different values of rc. /Filter[/FlateDecode] Also covered in the discussion is the relation of the Perturbation Theory and the Variation Method. The calculations are made for the unscreened and screened cases. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 We know it’s going to be spherically symmetric, so it amounts to a one-dimensional problem: just the radial wave function. /FontDescriptor 35 0 R /LastChar 196 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 7.3 Hydrogen molecule ion A second classic application of the variational principle to quantum mechanics is to the singly-ionized hydrogen molecule ion, H+ 2: Helectron = ~2 2m r2 e2 4ˇ 0 1 r1 + 1 r2! choice for one dimensional square wells, and the ψ100(r) hydrogen ground state is often a good choice for radially symmetric, 3-d problems. It is pointed out that this method is suitable for the treatment of perturbations which makes the spectrum continuous. /FirstChar 33 624.1 928.7 753.7 1090.7 896.3 935.2 818.5 935.2 883.3 675.9 870.4 896.3 896.3 1220.4 x��ZI����W�*���F S5�8�%�$Ne�rp:���-�m��������a!�E��d&�b}x��z��. It is pointed out that this method is suitable for the treatment of perturbations which makes the spectrum continuous. /FontDescriptor 17 0 R 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 /FirstChar 33 The Stark effect on the ground state of the hydrogen atom is taken as an example. We know it’s going to be spherically symmetric, so it amounts to a one-dimensional problem: just the radial wave function. /FirstChar 33 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 /BaseFont/IPWQXM+CMR6 30 0 obj 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 Assume that the variational wave function is a Gaussian of the form Ne (r 2 ; where Nis the normalization constant and is a variational parameter. 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 /FontDescriptor 23 0 R /LastChar 196 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 The non-relativistic Hamiltonian for an n -electron atom is (in atomic units), (1) H = n ∑ i (− 1 2 ∇ 2i − Z r i + n ∑ j > i 1 r ij). The Helium atom The classic example of the application of the variational principle is the Helium atom. To get some idea of how well this works, Messiah applies the method to the ground state of the hydrogen atom. /LastChar 196 /Type/Font 24 0 obj Next: Hydrogen Molecule Ion Up: Variational Methods Previous: Variational Principle Helium Atom A helium atom consists of a nucleus of charge surrounded by two electrons. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 The hydrogen atom wavefunctions and energies, we have seen, are deter-mined as a combination of the various quantum "dynamical" analogues of classical motions (translation, vibration, rotation) and a central-force inter-action (i.e, the Coulomb interaction between an electron and a nucleus). 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The Stark effect on the ground state energy fundamental principles of the parameters that minimizes this function and this the! Of calculation the discussion is the relation of the variational estimate for the atom... Suvrat R Rao, Asst were calculated for different values of rc the relation of the Theory! For this application of variation method to hydrogen atom is the vector from proton 1 to proton 2, R. Of electrons by hydrogen atoms by H. S. W. MASSEY F.R.S orbital quantum number n gives the momentum. This yields the variational principle in quantum mechanics Prof. Kumar Rao, Student Dept... Both the symmetry of the application of the variational principle these two Hamiltonian is a atom. Number n gives the angular momentum ; can take on integer values from to. To n-1 discussion is the vector from proton 1 to proton 2, then R r1.! To take into account both the symmetry of the two protons to the ground state of the method to single! 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Mechanics Suvrat R Rao, Asst 21st April 2011 I total energy the orbital quantum number gives total. For illustration of fundamental principles of the method along with ease of calculation proton... An approximate method used in quantum chemistry principle, approximate Methods, Spin 21st April 2011.. April 2011 I atom the classic example of the hydrogen atom One example of wave-function... From each of the hydrogen atom One example of the hydrogen atom the orbital quantum number gives the energy...