7.Approximate methods -the Variation MethodInafamouspaperpublishedin1929,Diracwrote:"The underlying physical laws necessary for the mathematical theory of ... the whole ofchemistry are thus completely known, and the difficulty is only that the exact application ofthese laws leads to equations much too complicated to be soluble. It therefore becomesdesirable that approximate practical methods of applying quantum mechanics should bedeveloped, which canlead to an explanation of themainfeatures of complex atomic systemswithout too much computation."With the help of computers, we can now solve many of the problems that Dirac consideredinsoluble in 1929. The most important tools for this purpose are the Variation Method andPerturbation Theory. Here we examine the first of these and show how it leads to the ideas thatwe use to understand chemical bonding
7. Approximate methods - the Variation Method In a famous paper published in 1929, Dirac wrote: "The underlying physical laws necessary for the mathematical theory of . the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation." With the help of computers, we can now solve many of the problems that Dirac considered insoluble in 1929. The most important tools for this purpose are the Variation Method and Perturbation Theory. Here we examine the first of these and show how it leads to the ideas that we use to understand chemical bonding
TheVariationPrincipleTheVariationMethod isbasedonthevariationprincipleThis asserts that if is an arbitrary wavefunction satisfying the boundary conditions for theproblem, then the expectation value of its energy is not less than the lowest eigenvalue of theHamiltonian.That is,E=(iv)zEo(i)(1)whereEisthelowesteigenvalueof H
The Variation Principle The Variation Method is based on the variation principle. This asserts that if 𝜓෨ is an arbitrary wavefunction satisfying the boundary conditions for the problem, then the expectation value of its energy is not less than the lowest eigenvalue of the Hamiltonian. That is, (1) where E0 is the lowest eigenvalue of H 0 H E E
ProofExpandbintermsofthenormalizedeigenfunctions of Hi=EcVkIf H is any linear Hermitian operator that represents a physically observable propertythentheeigenfunctionsofHformacompleteset.()=Ecrc,(wk/w.)(|HU)=Ecc, (y|Hy)=EcicouEcrc (klE,lVi)klkl=ZlcPEc'c,E,owH-Elc E
Proof Expand 𝜓෨ in terms of the normalized eigenfunctions of H k k k c * * 2 k l k l kl k l kl kl k k c c c c c If H is any linear Hermitian operator that represents a physically observable property, then the eigenfunctions of H form a complete set. * * * 2 k l k l kl k l k l l kl k l l kl kl k k k H c c H c c E c c E c E
so thatE_(H)_Z1cE()Z,cZ,lc'E Z,le'E。Z,e(Ex-E)E-E.≥0Z,lc,leZ,leif E is the ground-state energy. Note that E = E, only if all the Ck are zero for states withEk > Eo. To get the energy exactly right we have to get the wavefunction exactly rightHowever a good approximation to the wavefunction will yield a good approximation to theenergy.To arrive at good approximation to the ground state energy E, we try many trialvariation functions and look for the one that gives the lowest value of the variationalintegral
2 2 k k k k k H c E E c so that 2 2 2 0 0 0 2 2 2 0 k k k k k k k k k k k c E c E c E E E E c c c if 𝐸0 is the ground-state energy. Note that 𝐸෨ = 𝐸0 only if all the 𝑐𝑘 are zero for states with 𝐸𝑘 > 𝐸0 . To get the energy exactly right we have to get the wavefunction exactly right. However a good approximation to the wavefunction will yield a good approximation to the energy. To arrive at good approximation to the ground state energy E, we try many trial variation functions and look for the one that gives the lowest value of the variational integral
Variationprinciplefora particlein a boxSupposethat wedid notknowtheground-state wavefunctionfora particlein aboxKnowing that it has to be zero when x = O and x = a, we might try the wavefunction =x(a - x).For this wavefunction we find(|)=[~x2(a-x)’dx=α / 30d?h2ah(iHv)=-x(a02m 3dx(i/H/vi)E=(H)=10h2 / 2ma?(i)01Theexact energyfor the ground state in this case is h?/8ma2, so the approximate result ishigher than the exact one by a factor of 10/π2 = 1.013. The wavefunction is not correct,but it gives a good estimate of the energy
Variation principle for a particle in a box Suppose that we did not know the ground-state wavefunction for a particle in a box. Knowing that it has to be zero when 𝑥 = 0 and 𝑥 = 𝑎, we might try the wavefunction 𝜓෨ = 𝑥(𝑎 − 𝑥). For this wavefunction we find 2 2 5 0 2 2 2 3 2 0 ( ) d / 30 d ( ) ( )d 2 d 2 3 a a x a x x a a H x a x x a x x m x m 2 2 10 / 2 H E H ma The exact energy for the ground state in this case is ℎ 2/8𝑚𝑎2 , so the approximate result is higher than the exact one by a factor of 10/𝜋 2 = 1.013. The wavefunction is not correct, but it gives a good estimate of the energy. 0 l
Variation method for the harmonic oscillatorUsually we use a trial function that contains one or more adjustable parameters, and minimizethe energy with respect to the parameters. We can use the variation method in this way to findthe harmonic osillator ground state, using the trial function i = e-ax?k-amHy =一y+一2mV元/0h'αV元/a+[k-a'ty"Hyjidx2α2m2mAlso*dx=/元/a-[hαK40(i)4m
Variation method for the harmonic oscillator Usually we use a trial function that contains one or more adjustable parameters, and minimize the energy with respect to the parameters. We can use the variation method in this way to find the harmonic oscillator ground state, using the trial function 𝜓෨ = e − 1 2 𝛼𝑥 2 . 2 2 2 1 2 2 2 H x k m m 2 2 2 * 1 / d / 2 2 2 H x k m m 2 2 2 2 1 2 4 4 4 H k E k m m m ෨�� Also ∗ 𝜓෨ d𝑥 = 𝜋/𝛼
E_(H)Kh'αh'ααh(il)4m4α2mmTo find the lowest energy, we minimize with respect to a andα too bigeasily find that aα = Vkm/h, as before.The first term in E is the expectation value <T) of the kineticαtposmallenergy,while the second term is theexpectationvalue<V)ofthe potential energy. If α is large, we get a sharply peaked which has a low potential energy but a high kinetic energy. Ifα is small,thewavefunction is broad and varies slowlywithx, so <T) is small, but it extends into regions where thepotential energy is high
2 2 2 2 1 2 4 4 4 H k E k m m m To find the lowest energy, we minimize with respect to 𝛼 and easily find that 𝛼 = 𝑘𝑚/ℏ, as before. The first term in 𝐸෨ is the expectation value 𝑇 of the kinetic energy, while the second term is the expectation value 𝑉 of the potential energy. If 𝛼 is large, we get a sharply peaked 𝜓෨ which has a low potential energy but a high kinetic energy. If 𝛼 is small, the wave function is broad and varies slowly with x, so 𝑇 is small, but it extends into regions where the potential energy is high
LinearcombinationofatomicorbitalsUsually we choose a trial wavefunction that has one or more adjustable parameters in it, andchoose values for the parameters that minimize the energy. An very important type of trialfunctionisthelinear combinationofatomic orbitals(L.C.A.O.),whichwecanillustrateforthehydrogenmoleculeionThe wavefunction for an individual hydrogen atom is the ls orbital; for ahydrogenmoleculeionwetrythewavefunctionji=c.Sa+C,Sbwhere s.is the normalized ls orbital for atom a and s, for atom b, and c.and c, arenumerical coefficientsthatweshalladjusttominimizetheenergy.Thisfunctionbehaveslike s,near nucleus a, where s, is small, and like s, near nucleus b
Linear combination of atomic orbitals Usually we choose a trial wavefunction that has one or more adjustable parameters in it, and choose values for the parameters that minimize the energy. An very important type of trial function is the linear combination of atomic orbitals (L.C.A.O.), which we can illustrate for the hydrogen molecule ion. The wavefunction for an individual hydrogen atom is the 1s orbital; for a hydrogen molecule ion we try the wavefunction where sa is the normalized 1s orbital for atom a and sb for atom b, and ca and cb are numerical coefficients that we shall adjust to minimize the energy. This function behaves like sa near nucleus a, where sb is small, and like sb near nucleus b. a a b b c s c s
The energy Efor the L.C.A.O.trial function isE= (ilHli)_ [(c.s. +c,s,)H(c,s, cs,)dt(c +c)α+2cac,β()c? +cz +2c.c,S[(casa +c,s,)(casa +c,s,)dtα=[ s,Hs,dt=[ s,Hs,dtβ-{s,Hs,dt-{s,Hs.dS=[s.s,dtα istheenergyofthehydrogen lsorbital,somewhatmodifiedbecausetheHamiltonianisforthemolecule, notthe atom.β is theenergy ofthe overlapdensityinthefieldofthemolecule, it describes the strength of the bonding, and like aα it is negative. S is the overlapintegral,andforsimplicityweneglectit
The energy 𝐸෨ for the L.C.A.O. trial function is 2 2 2 2 d 2 d 2 a a b b a a b b a b a b a a b b a a b b a b a b H c s c s H c s c s c c c c E c s c s c s c s c c c c S d d d d d a a b b a b b a a b s Hs s Hs s Hs s Hs S s s 𝛼 is the energy of the hydrogen 1s orbital, somewhat modified because the Hamiltonian is for the molecule, not the atom. 𝛽 is the energy of the overlap density in the field of the molecule; it describes the strength of the bonding, and like 𝛼 it is negative. S is the overlap integral, and for simplicity we neglect it
E= (C+c)α+2c.c,βc +c +2c.c,SS~0(ca +c)E =(ca +c)α+ 2cacβdifferentiate with respect to CaaE2c,E+(c +c=2αc.+2βcbacaAt the minimum, aE/ac. = O, so this becomes(α- E)ca +βc, =0
2 2 2 2 2 2 a b a b a b a b c c c c E c c c c S 2 2 2 2 2 a b a b a b c c E c c c c 𝑆 ≈ 0 2 2 2 2 2 a a b a b a E c E c c c c c differentiate with respect to 𝑐𝑎 At the minimum, 𝜕𝐸෨/𝜕𝑐𝑎 = 0, so this becomes 0 E c c a b