Physical Chemistry CHEM 3510
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Date Created: 10/21/15
CHEM 3510 Fall 2006 Unit VI Approximation Methods in Quantum Mechanics A Introduction 1 Atomic units a b Are used in atomic and molecular calculations An advantage in their use is that the values expressed in atomic units are not affected by refinements in various constants me e h etc Unit of angular momentum h 475th mee El 1 a0 0529177 A Unit of length a0 named and denoted as 4 Unit of energy E named and denoted as 9 l6 28 h2 a 1 Eh 26255 kJmol 272116 eV 2 Quantum mechanical treatment of the He atom a The hydrogen atom serves as prototype for the treatment of more compleX atoms By solving the Schrodinger equation for the hydrogen atom 2 2 2hm 2 e wr6 ElV a 4711907 one obtains the allowed energies and the wavefunctions of the electron He atom is constituted from the nucleus and two electrons The wavefunction of the system will depend on El the position of the helium nucleus R El the positions of the two electrons r1 and r2 CHEM 3510 Fall 2006 e Schrodinger equation for the He atom 722 2 722 2 722 2 v v v Rrr 2M 2me 1 2me 2 M 1 2 2 2 2e 2e WR3rlar2 4780lR l ll 47Z3980R 1 2 47Z3980I391 1 2 EVRariarz El He atom is a threebody system and solving its Schrodinger equation is more complicated It is impossible to be solved exactly f Solving Schrodinger equation for the He atom El Consider the nucleus to be xed at the origin hz 2 2 V1 V2Vrlar2 me 2 2 2e 1 1 e Vrlar2 lrlar2 47I6 0 7391 7392 4780ll 1 l 2 EWF1F2 El This equation cannot be solved exactly El This is due to the existence of the El Without it one can apply the separation of variables technique g Schrodinger equation for the He atom represents a system that cannot be solved exactly Solving these types of equations requires approximate methods El Variational method El Perturbation theory CHEM 3510 Fall 2006 B Variational Method 1 Variational principle or Variation principle a Consider an arbitrary system for which the groundstate wavefunction and groundstate energy satisfy H 110 2 E0 110 1 A l VOH V1001 E0 l Woe10d Where drrepresents the volume element If one substitute 110 with any other function and calculate jf gjdr Mm then according to variational principle E 2 E0 Where E 2 E0 only if 110 E Variational Principle If an arbitrary wavefunction is used to calculate the energy then the calculated value is never less than the true ground state energy E0 El Variational principle says that one can calculate an upper bond to E0 by using any other function 2 Trial functions a One can choose the function called that depends of some arbitrary parameters a 8 called The energy calculated based on this trial function will also depend on these parameters Optimize the variational parameters a 8 7 to get the lowest groundstate energy therefore obtaining the best trial wavefunction CHEM 3510 Fall 2006 d Example of the ground state of the hydrogen atom 2 2 CI The Hamiltonian is 12 h d r2 e zmerz J dr 4780 2 El Choose the trial function r 0 Where 05 is a variational parameter El Determine Ea 12 2 2 4 3 2me 212807Z32 187383h4 El Optimize the expression of EOc with respect to 05 4 gt Emm 0424 167 80 mee4 0 compare to E0 0500 W 167 80 F CI The trial function with the optimal value of 05 3 My 2 L3 e 87r9r2 mg 3 7 m0 0 compare to 1113 3 Variational method for the He atom a Rewrite the Hamiltonian as 2 H HH1HH2 e i 4780 112 2 2 A h 2 2e 1 a HH J V 2me 471190 r b Neglect term so separation of variables is possible CHEM 3510 Fall 2006 One has HH 1ny r 6 j EjyH rj 451 2 get wand 13 Use a trial function 0 r1r2 111Sr111sr2 Calculate EZ 4 2 22 an 167 80 h 8 8 122 J 0H odz CI Eh is the atomic unit of energy called hartree Minimizing EZ gt Zmin gt Emin 28477Eh CI The value of 2mm can be interpreted as gt each electron partially screens the nucleus from the other 2 lt 2 El Compare with accurate calculated value 29037E11 El Experimental value 29033E1l 4 Variational method for a trial function obtained as a linear combination of functions a b C Consider a more compleX trial function N 5 ZCnfn clfl 02f2 nl El cl 62 coefficients are Example for N 2 5 clfl 02f2 jf gjdr Wm A 2 2 Cl H11 0102H12 0102H2102H22 2 2 I df 01S1120102S12 02522 El Hi and S17 are called The energy E01302 CHEM 3510 d Fall 2006 a HU 419195011 is called El Slj flfJdr is called El H lj H if H is Hermitian 2 2 011H11 2010217112 02H22 E01302 012511 20102512 0522 Minimize the energy with respect to the variational parameters 5E E20 3 61H11 E51102H12 E512 0 5E acz 0 I 01H12 ES1202H22 ES220 El Coefficients cl and 02 are nonzero if and only if H11 E511 H12 E512 0 H12 E512 H22 E522 o This is called El Solving the secular determinant H11 E511H22 E522 H12 E5122 0 3 H11H22 EH11322 H22311E2311322 HIZZ EH1223122 EZSIZZ 0 3 E2311322 122E17391223122 H11S22 H22311H11H22 HIZZ 0 o This is a secondorder equation called 0 The smallervalue solution is the variational approximation for the groundstate energy For the case of larger N than the determinant in N order H11ES11 H12ES12 HiN ESIN H12 ESIZ H22 Eszz 0 HiN ESIN H2NES2N HNN ESNN Once E is determined one can go back in determining coefficients 01 A trial function that depends linearly on the variational parameters leads to 90 CHEM 3510 Fall 2006 5 Slater orbitals a A more complex function can be used for trial functions N 2 Z cjfj j1 0quot2 f 2 6 J where function f j is function of few coefficients as well El Solving for the wavefunction becomes more demanding but algorithms are available b Slater filled the need for general and suitable trial functions that are not necessarily same as the hydrogen wavefunctions by introducing a set of orbitals called c Slater orbital are defined as Snlm r26 anrn le grylm 62 21 2 W Y m 6 are the spherical harmonics where N n is a normalization constant d Properties of Slater orbitals El Q zeta is arbitrary and is not necessarily equal to Zn as in hydrogenlike orbitals CI The radial part of the Slater orbitals El Snlm r6 is not orthogonal to Snrlm r6 El 11 can be also considered as a variational parameter and is optimized to get the lowest energy 91 CHEM 3510 Fall 2006 C Perturbation Theory 1 Description a Suppose one is unable to solve the Schrodinger equation 3111 2 E 11 for the system of interest but one can solve it for a similar system 0w0 E0l0 Where Ii 2 1310 HG El 310 is called El 311 is called b If the perturbation is small the solutions of Ii will be similar to those of 1170 c Anharmonic oscillator example 2 2 Hz h d lkx2lyx3ix4m Zydxz 2 6 24 D Hlt0gt d2 1 2 kx2 is the harmonic oscillator operator zJ dx 2 CI The solutions are known 1150 x E50 2 n 1 2h V El 191 2 17x3 1x4 is the perturbation 6 24 d Perturbation theory says that the wavefunction and the energy of the unperturbed system can be successively corrected 11 10 110 yZ E Em EltD E2 Cl yO is the El E 0 is the El 110 w2 are successive corrections El E0 E2 are successive corrections El A basic assumption is that those successive corrections become smaller Expressions are available for these terms 92 CHEM 3510 Fall 2006 e We will only work with E which is f E Em E is the g 11 yO 110 is the h E E 0 E 1 E 2 is the energy through secondorder perturbation theory 2 Application to Helium atom a Schrodinger equation for He atom 2 2 2km V12 VlF1J2 4278 rlriVFlirz e 0 l 2 2 e VF13F2 EVI lil z 47ZSOF1 l 2 b Unperturbed system 190 HH1HH2 2 2 A h 2 Where HH139 2 12 6 me 478071 0 Wm W1s l 91 1l1s7262 2 Em zzmee4 zzmee4 3272837221112 3272802h2ng c The perturbation is 1311 2 aim d The rstorder correction to the energy E0 2 5 2 mee4 5 th 8 l6n283h2 8 93 CHEM 3510 Fall 2006 The energy through rstorder perturbation theory E E0 EltD 122 122 22 2 22 22 2 2 8 8 El when Z 2 gt E 2750 Eh El variational result E 28477 Eh El experimental value E 29033 Eh CI The results do not look as good as variational method but second order perturbation theory gives E 29077 Eh El Considering that 1 Eh 26255 kJmol 005 Eh is a substantial value so higher level of corrections are needed for very accurate results Overall both variational method and perturbation theory can give reasonable results 94 CHEM 3510 Fall 2006 D Unit Review 1 Important Terminology atomic units interelectronic repulsion variational principle trial function variational parameters effective nuclear charge matrix elements Coulomb integral resonance integral secular determinant secular equation Slater orbitals Perturbation theory 95
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