Computational chemistry

Transcription

1 6 Computational hemisty The ental hallenge 6. The Hatee Fok fomalism 6. The Roothaan equations 6.3 Basis sets The fist appoah: semiempiial methods 6.4 The Hükel method evisited 6.5 Diffeential ovelap The seond appoah: ab initio methods 6.6 Configuation inteation 6.7 Many-body petubation theoy The thid appoah: density funtional theoy 6.8 The Kohn Sham equations 6.9 The exhange oelation enegy Cuent ahievements 6.0 Compaison of alulations and expeiments 6. Appliations to lage moleules I6. Impat on nanosiene: The stutues of nanopatiles I6. Impat on mediine: Moleula eognition and dug design Cheklist of key ideas Disussion questions Exeises Poblems In this hapte we extend the desiption of the eletoni stutue of moleules pesented in Chapte 5 by intoduing methods that haness the powe of omputes to alulate eletoni wavefuntions and enegies. These alulations ae among the most useful tools used by hemists fo the pedition of moleula stutue and eativity. The omputational methods we disuss handle the eleton eleton epulsion tem in the Shödinge equation in diffeent ways. One suh method, the Hatee Fok method, teats eleton eleton inteations in an aveage and appoximate way. This appoah typially equies the numeial evaluation of a lage numbe of integals. Semiempiial methods set these integals to zeo o to values detemined expeimentally. In ontast, ab initio methods attempt to evaluate the integals numeially, leading to a moe peise teatment of eleton eleton inteations. Configuation inteation and Mølle Plesset petubation theoy ae used to aount fo eleton oelation, the tendeny of eletons to avoid one anothe. Anothe omputational appoah, density funtional theoy, fouses on eleton pobability densities athe than on wavefuntions. The hapte onludes by ompaing esults fom diffeent eletoni stutue methods with expeimental data and by desibing some of the wide ange of hemial and physial popeties of moleules that an be omputed. The field of omputational hemisty, the use of omputes to pedit moleula stutue and eativity, has gown in the past few deades due to the temendous advanes in ompute hadwae and to the development of effiient softwae pakages. The latte ae now applied outinely to ompute moleula popeties in a wide vaiety of hemial appliations, inluding phamaeutials and dug design, atmosphei and envionmental hemisty, nanotehnology, and mateials siene. Many softwae pakages have sophistiated gaphial intefaes that pemit the visualization of esults. The matuation of the field of omputational hemisty was eognized by the awading of the 998 Nobel Pize in Chemisty to J.A. Pople and W. Kohn fo thei ontibutions to the development of omputational tehniques fo the eluidation of moleula stutue and eativity. The ental hallenge The goal of eletoni stutue alulations in omputational hemisty is the solution of the eletoni Shödinge = EΨ, whee E is the eletoni enegy and Ψ is the many-eleton wavefuntion, a funtion of the oodinates of all the eletons and the nulei. To make pogess, we invoke at the outset the Bon Oppenheime appoximation and the sepaation of eletoni and nulea motion (Chapte 5). The eletoni is

2 6 COMPUTATIONAL CHEMISTRY 73 e e n e = Ze I e i + m 4πε 4πε whee e the fist tem is the kineti enegy of the N e eletons; (6.a) the seond tem is the potential enegy of attation between eah eleton and eah of the N n nulei, with eleton i at a distane Ii fom nuleus I of hage Z I e; the final tem is the potential enegy of epulsion between two eletons sepaated by ij. The fato of in the final sum ensues that eah epulsion is ounted only one. The ombination e /4πε 0 ous thoughout omputational hemisty, and we denote it j 0. Then the hamiltonian beomes $ e = ZI i j0 + m e N i N e i We shall use the following labels: (6.b) Speies Label Numbe used Eletons i and j =,,... N e Nulei I = A, B,... N n Moleula obitals, ψ m = a, b,... Atomi obitals used to onstut the o =,,... N b moleula obitals (the basis ), χ Anothe geneal point is that the theme we develop in the sequene of illustations in this hapte is aimed at showing expliitly how to use the equations that we have pesented, and theeby give them a sense of eality. To do so, we shall take the simplest possible many-eleton moleule, dihydogen (H ). Some of the tehniques we intodue do not need to be applied to this simple moleule, but they seve to illustate them in a simple manne and intodue poblems that suessive setions show how to solve. One onsequene of hoosing to develop a stoy in elation to H, we have to onfess, is that not all the illustations ae atually as bief as we would wish; but we deided that it was moe impotant to show the details of eah little alulation than to adhee stitly to ou nomal use of the tem bief. N i N i N I N I 0 Ii Ii i j Ne j0 i j ij 0 ij The notation we use fo the desiption of H is shown in Fig. 6.. Fo this two-eleton (N e = ), two-nuleus (N n = ) moleule the hamiltonian is = ( + ) j me A B A B + j N 0 To keep the notation simple, we intodue the one-eleton opeato h i Eleton A Nuleus A $ = i j0 + m (6.) whih should be eognized as the hamiltonian fo eleton i in an H + moleule-ion. Then = h + h + e Ai Bi 0 A R AB Eleton Nuleus B Fig. 6. The notation used fo the desiption of moleula hydogen, intodued in the bief illustation peeding Setion 6. and used thoughout the text. (6.3) We see that the hamiltonian fo H is essentially that of eah eleton in an H + -like moleule-ion but with the addition of the eleton eleton epulsion tem. l It is hopeless to expet to find analytial solutions with a hamiltonian of the omplexity of that shown in eqn 6., even fo H, and the whole thust of omputational hemisty is to fomulate and implement numeial poedues that give eve moe eliable esults. 6. The Hatee Fok fomalism The eletoni wavefuntion of a many-eleton moleule is a funtion of the positions of all the eletons, Ψ(,,...). To fomulate one vey widely used appoximation, we build on the mateial in Chapte 5, whee we saw that in the MO desiption of H we supposed that eah eleton oupies an obital and that the oveall wavefuntion an be witten ψ( )ψ( )... Note that this obital appoximation is quite sevee and loses many of the details of the dependene of the wavefuntion on the elative loations of the eletons. We do the same hee, with two small hanges of notation. To simplify the appeaane of the expessions we wite ψ( )ψ( )...as ψ()ψ()....next, we suppose that eleton oupies a moleula obital ψ a with spin α, eleton oupies the same obital with spin β, and so on, and hene wite the many-eleton wavefuntion Ψ as the podut Ψ = ψ α a ()ψ β a ()....The ombination of a moleula obital and a spin funtion, suh as ψ α a (), is the spinobital α intodued in Setion 4.4; fo example, the spinobital ψ a should be intepeted as the podut of the spatial wavefuntion ψ a and the spin state α, so ψ α a () = ψ a ()α(), and likewise B B

3 74 6 COMPUTATIONAL CHEMISTRY fo the othe spinobitals. We shall onside only losed-shell moleules but the tehniques we desibe an be extended to open-shell moleules. A simple podut wavefuntion does not satisfy the Pauli piniple and hange sign unde the intehange of any pai of eletons (Setion 4.4). To ensue that the wavefuntion does satisfy the piniple, we modify it to a sum of all possible pemutations, using plus and minus signs appopiately: Ψ = ψ α a ()ψ β a ()...ψ β z (N e ) ψ α a ()ψ β a ()...ψ β z (N e ) +... (6.4) Thee ae N e! tems in this sum, and the entie sum an be epesented by the Slate deteminant (Setion 4.4): Ψ = N e! ψα a() ψβ a() ψβ z() α ψa() β ψa() β ψz() ψα( N ) ψβ( N ) ψβ( N ) (6.5a) The fato / N e! ensues that the wavefuntion is nomalized if the omponent moleula obitals ψ m ae nomalized. To save the tedium of witing out lage deteminants, the wavefuntion is nomally witten by using only its pinipal diagonal: Ψ = (/N e!) / ψ a α ()ψ a β ()...ψ z β (N e ) The Slate deteminant fo H (N e = ) is Ψ = ψ α a() ψβ a() ψ α() ψβ() = { ψ α( ) ψβ( ) ψβ( ) ψ α( )} a e a e z e = ψ () ψ ( ){ α() β( ) β( ) α( )} a a a a a a a a (6.5b) whee both eletons oupy the moleula wavefuntion ψ a. We should eognize the spin fato as that oesponding to a singlet state (eqn 4.3b, σ = (/ ){αβ βα}), so Ψ oesponds to two spin-paied eletons in ψ a. l Aoding to the vaiation piniple (Setion 5.5), the best fom of Ψ is the one that oesponds to the lowest ahievable enegy as the ψ ae vaied, that is, we need the wavefuntions ψ that will minimize the expetation value Ψ*@Ψ dτ. Beause the eletons inteat with one anothe, a vaiation in the fom of ψ a, fo instane, will affet what will be the best fom of all the othe ψs, so finding the best fom of the ψs is a fa fom tivial poblem. Howeve, D.R. Hatee and V. Fok showed that the optimum ψs eah satisfy an at fist sight vey simple set of equations: f ψ a () = ε a ψ a () (6.6) Fig. 6. A shemati intepetation of the physial intepetation of the Coulomb epulsion tem, eqn 6.7a. An eleton in obital ψ a expeienes epulsion fom an eleton in obital ψ m whee it has pobability density ψ m. whee f is alled the Fok opeato. This is the equation to solve to find ψ a ; thee ae analogous equations fo all the othe oupied obitals. This Shödinge-like equation has the fom we should expet (but its fomal deivation is quite involved). Thus, f has the following stutue: f = oe hamiltonian fo eleton (h ) + aveage Coulomb epulsion fom eletons, 3,... (V Coulomb ) + aveage oetion due to spin oelation (V Exhange ) = h + V Coulomb + V Exhange By the oe hamiltonian we mean the one-eleton hamiltonian h defined by eqn 6. and epesenting the enegy of eleton in the field of the nulei. The Coulomb epulsion fom all the othe eletons ontibutes a tem that ats as follows (Fig. 6.): J (6.7a) This integal epesents the epulsion expeiened by eleton in obital ψ a fom eleton in obital ψ m, whee it is distibuted with pobability density ψ m *ψ m. Thee ae two eletons in eah obital, so we an expet a total ontibution of the fom V whee the sum is ove all the oupied obitals, inluding obital a. You should be alet to the fat that ounting fo the obital with m = a is inoet, beause eleton inteats only with the seond eleton in the obital, not with itself. This eo will be oeted in a moment. The spin oelation tem takes into aount the fat that eletons of the same spin tend to avoid eah othe (Setion 4.4), whih edues the net Coulomb inteation between them. This ontibution has the following fom: K () ψ () = j () 0 ψ ψ *( ) ψ ( ) dτ m a a m m Coulomb ψ () = J () ψ () a m a m () ψ () = j () 0 ψ ψ *( ) ψ ( ) dτ m a m m a (6.7b)

4 6 COMPUTATIONAL CHEMISTRY 75 Fo a given eleton thee is only one eleton of the same spin in all the oupied obitals, so we an expet a total ontibution of the fom V Exhange ψ () = K () ψ () a m a m The negative sign eminds us that spin-oelation keeps eletons apat, and so edues thei lassial, Coulombi epulsion. By olleting tems, we aive at a speifi expession fo the effet of the Fok opeato: fψ () = hψ () + { J () K ()} ψ () a a m m a m (6.8) with the sum extending ove all the oupied obitals. Note that K a ()ψ a () = J a ()ψ a (), so the tem in the sum with m = a loses one of its J a, whih is the oetion that avoids the eleton epelling itself, whih we efeed to above. Equation 6.8 eveals a seond pinipal appoximation of the Hatee Fok fomalism (the fist being its dependene on the obital appoximation). Instead of eleton (o any othe eleton) esponding to the instantaneous positions of the othe eletons in the moleule though tems of the fom / j, it esponds to an aveaged loation of the othe eletons though integals of the kind that appea in eqn 6.7. When we look fo easons why the fomalism gives poo esults, this appoximation is a pinipal eason; it is addessed in Setion 6.6. Although eqn 6.6 is the equation we have to solve to find ψ a, eqn 6.7 eveals that it is neessay to know all the othe oupied wavefuntions in ode to set up the opeatos J and K and hene to find ψ a. To make pogess with this diffiulty, we an guess the initial fom of all the one-eleton wavefuntions, use them in the definition of the Coulomb and exhange opeatos, and solve the Hatee Fok equations. That poess is then ontinued using the newly found wavefuntions until eah yle of alulation leaves the enegies ε m and wavefuntions ψ m unhanged to within a hosen iteion. This is the oigin of the tem self-onsistent field (SCF) fo this type of poedue in geneal and of Hatee Fok selfonsistent field (HF-SCF) fo the appoah based on the obital appoximation. We ontinue with the H example. Aoding to eqn 6.6, the Hatee Fok equation fo ψ a is f ψ a () = ε a ψ a () with f ψ a () = h ψ a () + J a ()ψ a () K a ()ψ a () beause thee is only one tem in the sum (thee is only one oupied obital). In this expession The equation to solve is theefoe This equation fo ψ a must be solved self-onsistently (and numeially) beause the integal that govens the fom of ψ a equies us to know ψ a aleady. In the following examples we shall illustate some of the poedues that have been adopted. l 6. The Roothaan equations The diffiulty with the HF-SCF poedue lies in the numeial solution of the Hatee Fok equations, an oneous task even fo poweful omputes. As a esult, a modifiation of the tehnique was needed befoe the poedue ould be of use to hemists. We saw in Chapte 5 how moleula obitals ae onstuted as linea ombinations of atomi obitals. This simple appoah was adopted in 95 by C.C.J. Roothaan and G.G. Hall independently, who found a way to onvet the Hatee Fok equations fo the moleula obitals into equations fo the oeffiients that appea in the LCAO used to simulate the moleula obital. Thus, they wote (as we did in eqn 5.34) ψ = $ ψa() j0 + ψa() m + j0 ψ a() a a = a a ψ *( ) ψ ( ) dτ ε ψ () Nb χ m om o o= e A B (6.9) whee om ae unknown oeffiients and the χ o ae the atomi obitals (whih we take to be eal). Note that this appoximation is in addition to those undelying the Hatee Fok equations beause the basis is finite and so annot epodue the moleula obital exatly. The size of the basis set (N b ) is not neessaily the same as the numbe of atomi nulei in the moleule (N n ), beause we might use seveal atomi obitals on eah nuleus (suh as the fou s and p obitals of a abon atom). Fom N b basis funtions, we obtain N b linealy independent moleula obitals ψ. We show in Justifiation 6. that the use of a linea ombination like in eqn 6.9 leads to a set of simultaneous equations fo the oeffiients alled the Roothaan equations. These equations ae best summaized in matix fom by witing F = S e (6.0) whee F is the N b N b matix with elements Ja() ψa() = Ka() ψa() = j a() 0 ψ ψa*( ) ψa( )dτ F oo = χ o ()f χ o ()dτ (6.a)

5 76 6 COMPUTATIONAL CHEMISTRY S is the N b N b matix of ovelap integals: S oo = χ o ()χ o ()dτ (6.b) hoie. The two possible linea ombinations oesponding to eqn 6.9 ae ψ a = Aa χ A + Ba χ B ψ b = Ab χ A + Bb χ B and is an N b N b matix of all the oeffiients we have to find: a b N a b N = Na b Nb b NN b b (6.) The fist olumn is the set of oeffiients fo ψ a, the seond olumn fo ψ b, and so on. Finally, e is a diagonal matix of obital enegies ε a, ε b,...: ε a 0 0 ε = 0 εb εn b Justifiation 6. The Roothaan equations (6.d) To onstut the Roothaan equations we substitute the linea ombination of eqn 6.9 into eqn 6.6, whih gives f χ () = ε χ () Now multiply fom the left by χ o () and integate ove the oodinates of eleton : Nb That is, Nb Nb om o m om o o= o= F oo χ () f χ () d τ = ε om o o o= Nb Nb F = ε S oo om m oo om o= o= This expession has the fom of the matix equation in eqn 6.0. b b Nb Soo m om o o o= In this illustation we show how to set up the Roothaan equations fo H. To do so, we adopt a basis set of eal, nomalized funtions χ A and χ B, ented on nulei A and B, espetively. We an think of these funtions as Hs obitals on eah nuleus, but they ould be moe geneal than that, and in a late illustation we shall make a omputationally moe fiendly χ () χ () d τ so the matix is = and the ovelap matix S is The Fok matix is with S = χ A ()χ B ()dτ with F oo = χ o ()f χ o ()dτ We shall exploe the expliit fom of the elements of F in a late illustation; fo now, we just egad them as vaiable quantities. The Roothaan equations ae theefoe F F AA BA Aa Ba S = S S F = F F AA BA AA BA F F AB BB Ab Bb S S F F AB BB AB BB = S Aa Ba Ab Bb S = S l ANOTHER BRIEF ILLUSTRATION 0 l ε b In this ontinuation of the peeding illustation, we establish the simultaneous equations oesponding to the Roothaan equations we have just established. Afte multiplying out the maties onstuted in the peeding illustation, we obtain F + F F + F F F F + F AA Aa AB Ba AA Ab AB Bb BA Aa + BB Ba BA Ab BB Bb ε + Sε ε + Sε = εab a + SεaA a εbb b + Sε b a Aa a Ba b Ab b Bb On equating mathing elements, we obtain the following fou simultaneous equations: F AA Aa + F AB Ba = ε a Aa + Sε a Ba F BA Aa + F BB Ba = ε a Ba + Sε a Aa F AA Ab + F AB Bb = ε b Ab + Sε b Bb F BA Ab + F BB Bb = ε b Bb + Sε b Ab Thus, to find the oeffiients fo the moleula obital ψ a, we need to solve the fist and seond equations, whih we an wite as (F AA ε a ) Aa + (F AB Sε a ) Ba = 0 (F BA Sε a ) Aa + (F BB ε a ) Ba = 0 S Thee is a simila pai of equations (the thid and fouth) fo the oeffiients in ψ b. l Ab Aa Ba Ab Bb εa 0

6 This is a quadati equation fo the obital enegies ε, and may be solved by using the quadati fomula. Thus, if we summaize the equation as aε + bε + = 0, then With these enegies established and taking the lowe of the two enegies to be ε a sine ψ a is oupied in gound-state H, we an onstut the oeffiients by using the elation Aa F = F 6 COMPUTATIONAL CHEMISTRY 77 ε = b ± ( b 4 a ) / a AB AA Sεa ε a Ba Fig. 6.3 The iteation poedue fo a Hatee Fok selfonsistent field alulation. in onjuntion with the nomalization ondition Aa + Ba + Aa Ba S =. (Fo this homonulea diatomi moleule, thee is, of ouse, a muh simple method of aiving at Aa = Ba.) l If we wite the Roothaan equations as (F Se) = 0 we see that they ae simply a olletion of N b simultaneous equations fo the oeffiients. This point was demonstated expliitly in the peeding illustation. Theefoe, they have a solution only if F es =0 (6.) In piniple, we an find the obital enegies that ou in e by looking fo the oots of this seula equation and then using those enegies to find the oeffiients that make up the matix by solving the Roothaan equations. Thee is a ath, though: the elements of F depend on the oeffiients (though the pesene of J and K in the expession fo f ). Theefoe, we have to poeed iteatively: we guess an initial set of values fo, solve the seula equation fo the obital enegies, use them to solve the Roothaan equations fo, and ompae the esulting values with the ones we stated with. In geneal they will be diffeent, so we use those new values in anothe yle of alulation, and ontinue until onvegene has been ahieved (Fig. 6.3). The two simultaneous equations fo the oeffiients in ψ a obtained in the pevious illustation have a solution if FAA ε FAB Sε = 0 F Sε F ε BA BB The deteminant expands to give the following equation: (F AA ε)(f BB ε) (F AB Sε)(F BA Sε) = 0 On olleting tems, we aive at ( S )ε (F AA + F BB SF AB SF BA )ε + (F AA F BB F AB F BA ) = 0 The pinipal outstanding poblem is the fom of the elements of the Fok matix F and its dependene on the LCAO oeffiients. The expliit fom of F oo is F oo = χ o ()h χ o ()dτ + j () () 0 χ o χ o m( ) m( ) ψ ψ dτ dτ j0 χ o() ψ m() m( ) o ( ) ψ χ dτ dτ m m (6.3) whee the sums ae ove the oupied moleula obitals. The dependene of F on the oeffiients an now be seen to aise fom the pesene of the ψ m in the two integals, fo these moleula obitals depend on the oeffiients in thei LCAOs. At this point we ae eady to takle the matix elements that ou in the teatment of H, using the LCAOs set up in a pevious illustation. As we saw thee, we need the fou matix elements F AA, F AB, F BA, and F BB. We show hee how to evaluate F AA. Only one moleula obital is oupied (ψ a ), so eqn 6.3 beomes FAA = χa() hχa () dτ + j () () 0 χ A χ A a( ) a( ) ψ ψ d τ d τ j0 χ () a() A ψ a( ) ( ) ψ χ A d τ d τ

7 78 6 COMPUTATIONAL CHEMISTRY With ψ a = Aa χ A + Ba χ B, the seond integal on the ight is j0 χ A() χ A() a a ψ ( ) ψ ( ) d τ d τ = j0 a + a χ A() χ A() { A χ A( ) B χ B( )} = A aaaj0 χ A() χ A() A A d d χ ( ) χ ( ) τ τ Fom now on we shall use the notation ( AB CD) = j0 A( ) B() C( ) D() d d χ χ χ χ τ τ (6.4) Integals like this ae fixed thoughout the alulation beause they depend only on the hoie of basis, so they an be tabulated one and fo all and then used wheneve equied. Ou task late in this hapte will be to see how they ae evaluated. Fo the time being, we an teat them as onstants. In this notation, the integal we ae evaluating beomes j0 χ A() χ A() ψ a( ) ψ a( ) d τ d τ = Aa Aa (AA AA) + Aa Ba (AA BA) + Ba Ba (AA BB) (We have used (AA BA) = (AA AB).) Thee is a simila tem fo the thid integal, and oveall F AA = E A + Aa (AA AA) + Aa Ba (AA BA) + Ba {(AA BB) (AB BA)} whee { A aχa( ) + B aχb( )} dτdτ + A abaj0 A() A() A( ) B( ) d d + χ χ χ χ τ τ... E A = χ A ()h χ A ()dτ (6.5) is the enegy of an eleton in obital χ A based on nuleus A, taking into aount its inteation with both nulei. Simila expessions may be deived fo the othe thee matix elements of F. The uial point, though, is that we now see how F depends on the oeffiients that we ae tying to find. l Self-test 6. Constut the element F AB using the same basis. G F AB = χ A () h χ B () d τ + Aa ( BA AA) I J + AaBa{( 3 BA AB) ( AA BB)} + Ba( BA BB) L 6.3 Basis sets One of the poblems with moleula stutue alulations now beomes appaent. The basis funtions appeaing in eqn 6.4 may in geneal be ented on diffeent atomi nulei so (AB CD) is in geneal a so-alled fou-ente, two-eleton integal. If thee ae seveal dozen basis funtions used to build the one-eleton wavefuntions, thee will be tens of thousands of integals of this fom to evaluate (the numbe of integals ineases as N b 4 ). The effiient alulation of suh integals poses the geatest hallenge in an HF-SCF alulation but an be alleviated by a leve hoie of basis funtions. The simplest appoah is to use a minimal basis set, in whih one basis funtion is used to epesent eah of the obitals in an elementay valene theoy teatment of the moleule, that is, we inlude in the basis set one funtion eah fo H and He (to simulate a s obital), five funtions eah fo Li to Ne (fo the s, s, and thee p obitals), nine funtions eah fo Na to A, and so on. Fo example, a minimal basis set fo CH 4 onsists of nine funtions: fou basis funtions to epesent the fou Hs obitals, and one basis funtion eah fo the s, s, p x, p y, and p z obitals of abon. Unfotunately, minimal basis set alulations fequently yield esults that ae fa fom ageement with expeiment. Signifiant impovements in the ageement between eletoni stutue alulations and expeiment an often be ahieved by ineasing the numbe of basis set funtions. In a double-zeta (DZ) basis set, eah basis funtion in the minimal basis is eplaed by two funtions; in a tiple-zeta (TZ) basis set, by thee funtions. Fo example, a double-zeta basis fo H O onsists of fouteen funtions: a total of fou basis funtions to epesent the two Hs obitals, and two basis set funtions eah fo the s, s, p x, p y, and p z obitals of oxygen. In a split-valene (SV) basis set, eah inne-shell (oe) atomi obital is epesented by one basis set funtion and eah valene atomi obital by two basis set funtions; an SV alulation fo H O, fo instane, uses thiteen basis set funtions. Futhe impovements to the auay of eletoni stutue alulations an often be ahieved by inluding polaization funtions in the basis; these funtions epesent atomi obitals with highe values of the obital angula momentum quantum numbe l than onsideed in an elementay valene theoy teatment. Fo example, polaization funtions in a alulation fo CH 4 inlude basis funtions epesenting d obitals on abon o p obitals on hydogen. Polaization funtions often lead to impoved esults beause atomi obitals ae distoted (o polaized) by adjaent atoms when bonds fom in moleules. One of the ealiest hoies fo basis set funtions was that of Slate-type obitals (STO) ented on eah of the atomi nulei in the moleule and of the fom χ = N a e b Y lml (θ,φ) (6.6)

8 6 COMPUTATIONAL CHEMISTRY 79 N is a nomalization onstant, a and b ae (non-negative) paametes, Y lml is a spheial hamoni (Table 3.), and (,θ,φ) ae the spheial pola oodinates desibing the loation of the eleton elative to the atomi nuleus. Seveal suh basis funtions ae typially ented on eah atom, with eah basis funtion haateized by a unique set of values of a, b, l, and m l. The values of a and b geneally vay with the element and thee ae seveal ules fo assigning easonable values. Fo moleules ontaining hydogen, thee is an STO ented on eah poton with a = 0 and b = /a 0, whih simulates the oet behaviou of the s obital at the nuleus (see eqn 4.4). Howeve, using the STO basis set in HF-SCF alulations on moleules with thee o moe atoms equies the evaluation of so many two-eleton integals (AB CD) that the poedue beomes omputationally impatial. The intodution of Gaussian-type obitals (GTO) by S.F. Boys lagely oveame the poblem. Catesian Gaussian funtions ented on atomi nulei have the fom χ = Nx i y j z k e α (6.7) whee (x,y,z) ae the Catesian oodinates of the eleton at a distane fom the nuleus, (i,j,k) ae a set of non-negative Fig. 6.4 Contou plots fo Gaussian-type obitals. (a) s-type Gaussian, e ; (b) p-type Gaussian xe ; () d-type Gaussian, xye. integes, and α is a positive onstant. An s-type Gaussian has i = j = k = 0; a p-type Gaussian has i + j + k = ; a d-type Gaussian has i + j + k = and so on. Figue 6.4 shows ontou plots fo vaious Gaussian-type obitals. The advantage of GTOs is that the podut of two Gaussian funtions on diffeent entes is equivalent to a single Gaussian funtion loated at a point between the two entes (Fig. 6.5). Theefoe, two-eleton integals on thee and fou diffeent atomi entes an be edued to integals ove two diffeent entes, whih ae muh easie to evaluate numeially. Thee ae no fou-ente integals in H, but we an illustate the piniple by onsideing one of the two-ente integals that appea in the Fok matix and, to be definite, we onside ( AB AB) = j0 A( ) B() A( ) B() d d χ χ χ χ τ τ We hoose an s-type Gaussian basis and wite χ A () = Ne α R A χ B () = Ne α R B whee is the oodinate of eleton and R I is the oodinate of nuleus I. The podut of two suh Gaussians, one ented on A and one ented on B, fo eleton, is χ A ()χ B () = N e α R A e α R B = N e α{ R A + R B } By using the elation R = ( R) ( R) = + R R we an onfim that R A + R B = R + R 0 whee R 0 = (R A + R B ) is the midpoint of the moleule and R = R A R B is the bond length. Hene χ A ()χ B () = N e αr e α R 0 The podut χ A ()χ B () is the same, exept fo the index on. Theefoe, the two-ente, two-eleton integal (AB AB) edues to e = 4 α α R0 α 0 R 0 d τ d τ ( AB AB) N je R e This is a single-ente two-eleton integal, with both exponential funtions spheially symmetial Gaussians ented on the midpoint of the bond, and muh faste to evaluate than the oiginal two-ente integal. l Fig. 6.5 The podut of two Gaussian funtions on diffeent entes is itself a Gaussian funtion loated at a point between the two ontibuting Gaussians. The sale of the podut has been ineased elative to that of its two omponents. Some of the basis sets that employ Gaussian funtions and ae ommonly used in eletoni stutue alulations ae given in Table 6.. An STO-NG basis is a minimal basis set in whih eah basis funtion is itself a linea ombination of N Gaussians; the STO in the name of the basis eflets the fat that eah linea

9 80 6 COMPUTATIONAL CHEMISTRY Table 6. Basis set designations and example basis sets fo H O Geneal basis Example basis Basis funtions STO-NG STO-3G Fo eah O s, s, p x, p y, p z and H s obital: One funtion, a linea ombination of 3 Gaussians m-npg 6-3G Fo O s obital: One linea ombination of 6 Gaussians Fo eah O s, p x, p y, p z and H s obital: funtions: One Gaussian funtion One linea ombination of 3 Gaussians m-npg* 6-3G* 6-3G plus d-type polaization funtions on O m-npg** 6-3G** 6-3G* plus p-type polaization funtions on eah H m-npqg 6-3G 6-3G plus an additional Gaussian fo eah O s, p x, p y, p z and H s obital m-npq+g 6-3+G 6-3G plus diffuse s- and p-type Gaussians on O m-npq++g 6-3++G 6-3+G plus diffuse Gaussians on eah H m-npq+g* 6-3+G* 6-3+G plus d-type polaization funtions on O m-npq+g** 6-3+G** 6-3+G* plus p-type polaization funtions on eah H ombination is hosen by a least-squaes fit to a Slate-type funtion. An m-npg basis is a split-valene basis set in whih eah oe atomi obital is epesented by one funtion (a linea ombination of m Gaussians) and eah valene obital is epesented by two basis funtions, one a linea ombination of n Gaussians and the othe of p Gaussian funtions. The addition of d-type polaization funtions fo non-hydogen atoms to the m-npg basis yields an m-npg* basis; futhe addition of p-type polaization funtions fo hydogen atoms esults in an m- npg** basis set. In an m-npqg basis, eah valene atomi obital is epesented by thee basis funtions, linea ombinations of n, p, and q Gaussians, espetively. Addition of diffuse (small α- valued, eqn 6.7) s- and p-type Gaussians on non-hydogen atoms esults in an m-npq+g basis set; additional diffuse funtions to hydogen, m-npq++g. A onsideable amount of wok has gone into the development of effiient basis sets and this is still an ative aea of eseah. We have aived at the point whee we an see that the Hatee Fok appoah, oupled with the use of basis set funtions, equies the evaluation of a lage numbe of integals. Thee ae two appoahes ommonly taken at this point. In semiempiial methods, the integals enounteed ae eithe set to zeo o estimated fom expeimental data. In ab initio methods, an attempt is made to evaluate the integals numeially, using as input only the values of fundamental onstants and atomi numbes of the atoms pesent in the moleule. The fist appoah: semiempiial methods In semiempiial methods, many of the integals that ou in a alulation ae estimated by appealing to spetosopi data o physial popeties suh as ionization enegies, o by using a seies of ules to set etain integals equal to zeo. These methods ae applied outinely to moleules ontaining lage numbes of atoms beause of thei omputational speed but thee is often a saifie in the auay of the esults. 6.4 The Hükel method evisited Semiempiial methods wee fist developed fo onjugated π systems, the most famous semiempiial poedue being Hükel moleula obital theoy (HMO theoy, Setion 5.6). The initial assumption of HMO theoy is the sepaate teatment of π and σ eletons, whih is justified by the diffeent enegies and symmeties of the obitals. The seula deteminant, fom whih the π-obital enegies and wavefuntions ae obtained, has a fom simila to that of eqn 6. and is witten in tems of ovelap integals and hamiltonian matix elements. The ovelap integals ae set to 0 o, the diagonal hamiltonian matix elements ae set to a paamete α, and off-diagonal elements eithe to 0 o the paamete β. The HMO appoah is useful fo qualitative, athe than quantitative, disussions of onjugated π systems beause it teats epulsions between eletons vey pooly. Hee we etun to the thid illustation of Setion 6. and set S = 0. The diagonal Fok matix elements ae set equal to α (that is, we set F AA = F BB = α), and the off-diagonal elements ae set equal to β (that is, we set F AB = F BA = β). Note that the dependene of these integals on the oeffiients is swept aside, so we do not have to wok towads self-onsisteny. The quadati equation fo the enegies ( S )ε (F AA + F BB SF AB SF BA )ε + (F AA F BB F AB F BA ) = 0 beomes simply ε αε + α β = 0 and the oots ae ε = α ± β, exatly as we found in Setion 5.6. l

10 6 COMPUTATIONAL CHEMISTRY Diffeential ovelap In the seond most pimitive and sevee appoah, alled omplete neglet of diffeential ovelap (CNDO), all two-eleton integals of the fom (AB CD) ae set to zeo unless χ A and χ B ae the same, and likewise fo χ C and χ D. That is, only integals of the fom (AA CC) suvive and they ae often taken to be paametes with values adjusted until the alulated enegies ae in ageement with expeiment. The oigin of the tem diffeential ovelap is that what we nomally take to be a measue of ovelap is the integal χ A χ B dτ. The diffeential of an integal of a funtion is the funtion itself, so in this sense the diffeential ovelap is the podut χ A χ B. The impliation is that we then simply ompae obitals: if they ae the same, the integal is etained; if diffeent, it is disaded. The expession fo F AA deived in the final illustation in Setion 6. is F AA = E A + Aa (AA AA) + Aa Ba (AA BA) + Ba {(AA BB) (AB BA)} The last integal has the fom ( AB BA) = j0 χ A( ) χ B() B( ) A() d d χ χ τ τ The diffeential ovelap tem χ A ()χ B () is set equal to zeo, so in the CNDO appoximation the integal is set equal to zeo. The same is tue of the integal (AA BA). It follows that we wite F AA E A + Aa (AA AA) + Ba (AA BB) and identify the suviving two two-eleton integals as empiial paametes. l Self-test 6. Apply the CNDO appoximation to F AB fo the same system. [F AB = χ A ()h χ B ()dτ Aa Bb (AA BB)] Moe eent semiempiial methods make less daonian deisions about whih integals ae to be ignoed, but they ae all desendants of the ealy CNDO tehnique. Wheeas CNDO sets integals of the fom (AB AB) to zeo fo all diffeent χ A and χ B, intemediate neglet of diffeential ovelap (INDO) does not neglet the (AB AB) fo whih diffeent basis funtions χ A and χ B ae ented on the same nuleus. Beause these integals ae impotant fo explaining enegy diffeenes between tems oesponding to the same eletoni onfiguation, INDO is muh pefeed ove CNDO fo spetosopi investigations. A still less sevee appoximation is neglet of diatomi diffeential ovelap (NDDO) in whih (AB CD) is negleted only when χ A and χ B ae ented on diffeent nulei o when χ C and χ D ae ented on diffeent nulei. Thee ae othe semiempiial methods, with names suh as modified intemediate neglet of diffeential ovelap (MINDO), modified neglet of diffeential ovelap (MNDO), Austin model (AM), PM3, and paiwise distane dieted Gaussian (PDDG). In eah ase, the values of integals ae eithe set to zeo o set to paametes with values that have been detemined by attempting to optimize ageement with expeiment, suh as measued values of enthalpies of fomation, dipole moments, and ionization enegies. MINDO is useful fo the study of hydoabons; it tends to give moe auate omputed esults than MNDO but it gives poo esults fo systems with hydogen bonds. AM, PM3, and PDDG ae impoved vesions of MNDO. The seond appoah: ab initio methods In ab initio methods, the two-eleton integals ae evaluated numeially. Howeve, even fo small moleules, Hatee Fok alulations with lage basis sets and effiient and auate alulation of two-eleton integals an give vey poo esults beause they ae ooted in the obital appoximation and the aveage effet of the othe eletons on the eleton of inteest. Thus, the tue wavefuntion fo H is a funtion of the fom Ψ(, ), with a ompliated behaviou as and vay and pehaps appoah one anothe. This omplexity is lost when we wite the wavefuntion as a simple podut of two funtions, ψ( )ψ( ) and teat eah eleton as moving in the aveage field of the othe eletons. That is, the appoximations of the Hatee Fok method imply that no attempt is made to take into aount eleton oelation, the tendeny of eletons to stay apat in ode to minimize thei mutual epulsion. Most moden wok in eletoni stutue, suh as the appoahes disussed in the following two setions as well as moe sophistiated appoahes that ae beyond the sope of this text, ties to take eleton oelation into aount. 6.6 Configuation inteation When we wok though the fomalism desibed so fa using a basis set of N b obitals, we geneate N b moleula obitals. Howeve, if thee ae N e eletons to aommodate, in the gound state only N e of these N b obitals ae oupied, leaving N b N e so-alled vitual obitals unoupied. The gound state is Ψ 0 = (/N e!) / ψ a α ()ψ a β ()ψ b α (3)ψ b β (4)...ψ u β (N e )

11 8 6 COMPUTATIONAL CHEMISTRY whee ψ u is the HOMO (Setion 5.6). We an envisage tansfeing an eleton fom an oupied obital to a vitual obital ψ v, and foming the oesponding singly exited deteminant, suh as Ψ = (/N e!) / ψ a α ()ψ a β ()ψ b α (3)ψ v β (4)...ψ u β (N e ) Hee a β eleton, eleton 4, has been pomoted fom ψ b into ψ v, but thee ae many othe possible hoies. We an also envisage doubly exited deteminants, and so on. Eah of the Slate deteminants onstuted in this way is alled a onfiguation state funtion (CSF). Now we ome to the point of intoduing these CSFs. In 959 P.-O. Löwdin poved that the exat wavefuntion (within the Bon Oppenheime appoximation) an be expessed as a linea ombination of CSFs found fom the exat solution of the Hatee Fok equations: Ψ = C 0 Ψ 0 ( ) + C Ψ ( ) + C Ψ ( ) +... (6.8) The inlusion of CSFs to impove the wavefuntion in this way is alled onfiguation inteation (CI). Configuation inteation an, at least in piniple, yield the exat gound-state wavefuntion and enegy and thus aounts fo the eleton oelation negleted in Hatee Fok methods. Howeve, the wavefuntion and enegy ae exat only if an infinite numbe of CSFs ae used in the expansion in eqn 6.8; in patie, we ae esigned to using a finite numbe of CSFs. We an begin to appeiate why CI impoves the wavefuntion of a moleule by onsideing H again. We saw in the fist illustation in Setion 6. that, afte expanding the Slate deteminant, the gound state is Ψ 0 = ψ a ()ψ a ()σ (,) whee σ (,) is the singlet spin state wavefuntion. We also know that if we use a minimal basis set and ignoe ovelap, we an wite ψ a = (/ ){χ A + χ B }. Theefoe Ψ 0 = {χ A () + χ B ()}{χ A () + χ B ()}σ (,) = {χ A ()χ A () + χ A ()χ B () + χ B ()χ A () + χ B ()χ B ()}σ (,) We an see a defiieny in this wavefuntion: thee ae equal pobabilities of finding both eletons on A (the fist tem) o on B (the fouth tem) as thee ae fo finding one eleton on A and the othe on B (the seond and thid tems). That is, eleton oelation has not been taken into aount and we an expet the alulated enegy to be too high. Fom two basis funtions we an onstut two moleula obitals: we denote the seond one ψ b = (/ ){χ A χ B }. We need not onside the singly exited deteminant onstuted by moving one eleton fom ψ a to ψ b beause it will be of ungeade symmety and theefoe not ontibute to the geade gound state of dihydogen. A doubly exited deteminant based on ψ b would be Ψ = ψ b ()ψ b ()σ (,) = {χ A () χ B ()}{χ A () χ B ()}σ (,) = {χ A ()χ A () χ A ()χ B () χ B ()χ A () + χ B ()χ B ()}σ (,) If we wee simply to subtat one CSF fom the othe, the oute tems would anel and we would be left with Ψ 0 Ψ = {χ A ()χ B () + χ B ()χ A ()}σ (,) Aoding to this wavefuntion, the two eletons will neve be found on the same atom: we have oveompensated fo eleton onfiguation. The obvious middle-gound is to fom the linea ombination Ψ = C 0 Ψ 0 + C Ψ and look fo the values of the oeffiients that minimize the enegy. l The illustation shows that even a limited amount of CI an intodue some eleton oelation; full CI using obitals built fom a finite basis and allowing fo all possible exitations will take eleton oelation into aount moe fully. The optimum poedue, using obitals that fom an infinite basis and allowing all exitations, is omputationally impatial. The optimum expansion oeffiients in eqn 6.8 ae found by using the vaiation piniple; as in Justifiation 6. fo the Hatee Fok method, appliation of the vaiation piniple fo CI esults in a set of simultaneous equations fo the expansion oeffiients. If we take the linea ombination Ψ = C 0 Ψ 0 + C Ψ, the usual poedue fo the vaiation method (Setion 5.5) leads to the seula equation H ES =0, fom whih we an find the impoved enegy. Speifially: H = H H S = S S S S H H and the seula equation we must solve to find E is (note that S 0 = S 0 and that H 0 = H 0 due to hemitiity) H ES H ES H ES H ES S MN = Ψ Ψ dτ M N dτ H MN = Ψ dτ M N dτ = ( H ES )( H ES ) ( H ES ) = whih is easily eaanged into a quadati equation fo E. As usual, the poblem boils down to an evaluation of vaious integals that appea in the matix elements.

12 6 COMPUTATIONAL CHEMISTRY 83 The moleula obitals ψ a and ψ b ae othogonal, so S is diagonal and, povided ψ a and ψ b ae nomalized, S 00 = S =. To evaluate the hamiltonian matix elements, we fist wite the hamiltonian as in eqn 6.3 (@ = h + h + j 0 / ), whee h and h ae the oe hamiltonians fo eletons and, espetively, and so j 0 H00 = Ψ 0 h + h + 0 Ψ dτ dτ The fist tem in this integal (noting that the spin states ae nomalized) is: Ψ 0 h Ψ 0 dτ dτ = ψ a ()ψ a ()h ψ a ()ψ a ()dτ dτ Similaly, = ψ a ()h ψ a ()dτ Ψ 0 h Ψ 0 dτ dτ = ψ a ()h ψ a ()dτ Fo the eleton eleton epulsion tem, using the notation of eqn 6.4, j0 Ψ0 Ψ0dτdτ= j0 ψ a() ψa() ψa( ) ψa( )dτdτ = 4 Aa (AA AA) + 3 Aa Ba (AA AB) Ba (BB BB) Expessions of a simila kind an be developed fo the othe thee elements of H, so the optimum enegy an be found by substituting the alulated values of the oeffiients and the integals into the expession fo the oots of the quadati equation fo E. The oeffiients in the CI expession fo Ψ an then be found in the nomal way by using the lowest value of E and solving the seula equations. l 6.7 Many-body petubation theoy The appliation of petubation theoy to a moleula system of inteating eletons and nulei is alled many-body petubation theoy. Reall fom disussions of petubation theoy in Chapte (see eqn.3) that the hamiltonian is expessed as a sum of a simple, model (0), and a (). Beause we wish to find the oelation enegy, a natual hoie fo the model hamiltonian ae the Fok opeatos of the HF-SCF method and fo the petubation we take the diffeene between the Fok opeatos and the tue many-eleton hamiltonian (eqn 6.). That (0) () () 0 = f i (6.9) N e i= Beause the oe hamiltonian in the Fok opeato in eqn 6.8 anels the one-eleton tems in the full hamiltonian, the petubation is the diffeene between the instantaneous inteation between the eletons (the thid tem in eqn 6.) and the aveage inteation (as epesented by the opeatos J and K in the Fok opeato). Thus, fo () j () = 0 { Jm() Km()} (6.0) whee the fist sum (the tue inteation) is ove all the eletons othe than eleton itself and the seond sum (the aveage inteation) is ove all the oupied obitals. This hoie was fist made by C. Mølle and M.S. Plesset in 934 and the method is alled Mølle Plesset petubation theoy (MPPT). Appliations of MPPT to moleula systems wee not undetaken until the 970s and the ise of suffiient omputing powe. As usual in petubation theoy, the tue wavefuntion is witten as a sum of the eigenfuntion of the model hamiltonian and highe-ode oetion tems. The oelation enegy, the diffeene between the tue enegy and the HF enegy, is given by enegy oetions that ae seond ode and highe. If we suppose that the tue wavefuntion of the system is given by a sum of CSFs like that in eqn 6.8, then (see eqn.35) E (6.) Aoding to Billouin s theoem, only doubly exited Slate deteminants have () matix elements and hene only they make a ontibution to E 0 (). The identifiation of the seond-ode enegy oetion with the oelation enegy is the basis of the MPPT method denoted MP. The extension of MPPT to inlude thid- and fouth-ode enegy oetions ae denoted MP3 and MP4, espetively. Aoding to Billouin s theoem, and fo ou simple model of H built fom two basis obitals, we wite Ψ = C 0 Ψ 0 + C Ψ with Ψ 0 = ψ a ()ψ a ()σ (,) Ψ = ψ b ()ψ b ()σ (,) The only matix element we need fo the sum in eqn 6. is i i m Ψ () Ψ 0 dτ () () E E 0 ( ) = M 0 0 M 0 0 Ψ@ () Ψ0dτdτ = j0 ψ () ( ) b ψb ψa() ψa( ) dτdτ All the integals ove tems based on J and K ae zeo beause these ae one-eleton opeatos and so eithe ψ a () o ψ a () is left unhanged and its othogonality to ψ b ensues that the integal vanishes. We now expand eah moleula obital in tems of the basis funtions χ A and χ B, and obtain Ψ 0 dτ dτ = Ab Aa (AA AA) + Ab Bb Aa (BA AA) Bb Ab (BB BB)

13 84 6 COMPUTATIONAL CHEMISTRY If we ignoe ovelap the oeffiients ae all equal to ±/, and if we use symmeties like (AA AB) = (AA BA) and (AA AB) = (BB BA), this expession simplifies to Ψ 0 dτ dτ = {(AA AA) (AA BB)} It follows that the seond-ode estimate of the oelation enegy is E ( ) 0 = 4 {( AA AA) ( AA BB)} E () 0 () 0 0 E {( AA AA) ( AA BB)} = 8( ε ε ) The tem (AA AA) (AA BB) is the diffeene in epulsion enegy between both eletons being onfined to one atom and eah being on a diffeent atom. l The thid appoah: density funtional theoy a b A tehnique that has gained onsideable gound in eent yeas to beome one of the most widely used poedues fo the alulation of moleula stutue is density funtional theoy (DFT). Its advantages inlude less demanding omputational effot, less ompute time, and in some ases, patiulaly fo d-metal omplexes bette ageement with expeimental values than is obtained fom Hatee Fok based methods. 6.8 The Kohn Sham equations The ental fous of DFT is not the wavefuntion but the eleton pobability density, ρ (Setion.5). The funtional pat of the name omes fom the fat that the enegy of the moleule is a funtion of the eleton density and the eleton density is itself a funtion of the positions of the eletons, ρ(). In mathematis a funtion of a funtion is alled a funtional, and in this speifi ase we wite the enegy as the funtional E[ρ]. We have enounteed a funtional befoe but did not use this teminology: the expetation value of the hamiltonian is the enegy expessed as a funtional of the wavefuntion, fo a single value of the enegy, E[ψ], is assoiated with eah funtion ψ. An impotant point to note is that beause E[ψ] is an integal of ψhψ ove all spae, it has ontibutions fom the whole ange of values of ψ. Simply fom the stutue of the hamiltonian in eqn 6. we an suspet that the enegy of a moleule an be expessed as ontibutions fom the kineti enegy, the eleton nuleus inteation, and the eleton eleton inteation. The fist two ontibutions depend on the eleton density distibution. The eleton eleton inteation is likely to depend on the same quantity, but we have to be pepaed fo thee to be a modifiation of the lassial eleton eleton inteation due to eleton exhange (the ontibution whih in Hatee Fok theoy is expessed by K). That the exhange ontibution an be expessed in tems of the eleton density is not at all obvious, but in 964 P. Hohenbeg and W. Kohn wee able to pove that the exat gound-state enegy of an N e -eleton moleule is uniquely detemined by the eleton pobability density. They showed that it is possible to wite E[ρ] = E Classial [ρ] + E XC [ρ] (6.) whee E Classial [ρ] is the sum of the ontibutions of kineti enegy, eleton nuleus inteations, and the lassial eleton eleton potential enegy, and E XC [ρ] is the exhange oelation enegy. This tem takes into aount all the non-lassial eleton eleton effets due to spin and applies small oetions to the kineti enegy pat of E Classial that aise fom eleton eleton inteations. The Hohenbeg Kohn theoem guaantees the existene of E XC [ρ] but like so many existene theoems in mathematis gives no lue about how it should be alulated. The fist step in the implementation of this appoah is to alulate the eleton density. The elevant equations wee dedued by Kohn and L.J. Sham in 965, who showed that ρ an be expessed as a ontibution fom eah eleton pesent in the moleule, and witten ρ() = ψi() (6.3) ψ i is alled a Kohn Sham obital and is a solution of the Kohn Sham equation, whih losely esembles the fom of the Shödinge equation (on whih it is based). Fo a two-eleton system, ρ() hψ i() + j0 dτψ i() + V ψi() = εψ i i (6.4) XC () The fist tem is the usual oe tem, the seond tem is the lassial inteation between eleton and eleton, and the thid tem takes exhange effets into aount and is alled the exhange oelation potential. The ε i ae the Kohn Sham obital enegies. 6.9 The exhange oelation enegy The exhange oelation potential plays a ental ole in DFT and an be alulated one we know the exhange oelation enegy E XC [ρ] by foming the following funtional deivative : V XC Ne i= EXC[ ] () = δ ρ δρ (6.5)

14 6 COMPUTATIONAL CHEMISTRY 85 A funtional deivative is defined like an odinay deivative, but we have to emembe that E XC [ρ] is a quantity that gets its value fom the entie ange of values of ρ(), not just fom a single point. Thus, when undegoes a small hange d, the density hanges by δρ to ρ( + d) at eah point and E XC [ρ] undegoes a hange that is the sum (integal) of all suh hanges: δe XC δexc[ ρ] [ ρ] = δρd = VXC() δρd δρ Note that V XC is an odinay funtion of, not a funtional: it is the loal ontibution to the integal that defines the global dependene of E XC [ρ] on δρ thoughout the ange of integation. The geatest hallenge in density funtional theoy is to find an auate expession fo the exhange oelation enegy. One widely used but appoximate fom fo E XC [ρ] is based on the model of a unifom eleton gas, a hypothetial eletially neutal system in whih eletons move in a spae of ontinuous and unifom distibution of positive hage. Fo a unifom eleton gas, the exhange oelation enegy an be witten as the sum of an exhange ontibution and a oelation ontibution. The latte is a ompliated funtional that is beyond the sope of this hapte; we ignoe it hee. Then the exhange oelation enegy is E XC [ρ] = Aρ 4/3 d with A = (9/8)(3/π) / j 0 When the density hanges fom ρ() to ρ() +δρ() at eah point (Fig. 6.6), the funtional hanges fom E XC [ρ] to E XC [ρ +δρ]: The integand an be expanded in a Taylo seies (Mathematial bakgound ) and, disading tems of ode δρ and highe, we obtain: E XC [ρ +δρ] = (Aρ 4/ Aρ /3 δρ)d = E XC [ρ] Aρ /3 δρd Theefoe, the diffeential δe XC of the funtional (the diffeene E XC [ρ +δρ] E XC [ρ] that depends linealy on δρ) is δe XC [ρ] = 4 3 Aρ /3 δρd and theefoe V XC () = 4 3 Aρ() /3 = 3 (3/π) /3 j 0 ρ() /3 (6.6) l Self-test 6.3 Find the exhange oelation potential if the exhange oelation enegy is given by E XC [ρ] = Bρ d. [V XC () = Bρ()] The Kohn Sham equations must be solved iteatively and self-onsistently (Fig. 6.7). Fist, we guess the eleton density; it is ommon to use a supeposition of atomi eleton pobability densities. Seond, the exhange oelation potential is alulated by assuming an appoximate fom of the dependene of the exhange oelation enegy on the eleton density and evaluating the funtional deivative. Next, the Kohn Sham equations ae solved to obtain an initial set of Kohn Sham E XC [ρ +δρ] = A(ρ +δρ) 4/3 d Contibution to enegy E[ ] E[ ] ( ) ( ) ( ) Loation Fig. 6.6 The hange in the exhange oelation enegy funtional fom E XC [ρ] to E XC [ρ +δρ] (the aea unde eah uve) as the density hanges fom ρ to ρ +δρ at eah point. Assume fom fo exhange oelation enegy, [ ] E XC Evaluate exhange oelation potential, V XC Kohn Sham equations, eqn 6.4 Kohn Sham obitals, Fomulate tial density, i Eletoni enegy, E, eqn 6. Yes No Convegene? Eleton density,, eqn 6.3 Fig. 6.7 The iteation poedue fo solving the Kohn Sham equations in density funtional theoy.

15 86 6 COMPUTATIONAL CHEMISTRY obitals. This set of obitals is used to obtain a bette appoximation to the eleton pobability density (fom eqn 6.3) and the poess is epeated until the density emains onstant to within some speified toleane. The eletoni enegy is then omputed by using eqn 6.. As is the ase fo the Hatee Fok one-eleton wavefuntions, the Kohn Sham obitals an be expanded using a set of basis funtions; solving eqn 6.4 then amounts to finding the oeffiients in the expansion. Vaious basis funtions, inluding Slate-type and Gaussian-type obitals, an be used. Wheeas Hatee Fok methods have omputational times that sale as N b 4, DFT methods sale as N b 3. Theefoe, DFT methods ae omputationally moe effiient, though not neessaily moe auate, than HF methods. In applying DFT to moleula hydogen, we begin by assuming that the eleton density is a sum of atomi eleton densities aising fom the pesene of eletons in the atomi obitals χ A and χ B (whih may be STOs o GTOs) and wite ρ() = χ A + χ B fo eah eleton. Fo the exhange oelation enegy E XC we use the fom appopiate to a unifom eleton gas and the oesponding exhange oelation potential deived in the pevious illustation (eqn 6.6). The Kohn Sham obital fo the moleule is a solution of 3 h + j0 3 j ρ ( ) d ( / π) ρ( ) We inset the ρ( ) and ρ( ) we have assumed and solve this equation numeially fo ψ. One we have that obital, we eplae ou oiginal guess at the eleton density by ρ() = ψ (). This density is then substituted bak into the Kohn Sham equation to obtain an impoved funtion ψ () and the poess epeated until the density and exhange oelation enegy ae unhanged to within a speified toleane on suessive iteations. When onvegene of the iteations has been ahieved, the eletoni enegy (eqn 6.) is alulated fom E[ ρ] = ψ( ) hψ() d+ j0 ρ( ) ρ( ) dd 9 8 ( 3/ π) 3 / j ( ) 43 / 0 ρ d 3 / 3 / 0 whee the fist tem is the sum of the enegies of the two eletons in the field of the two nulei, the seond tem is the eleton eleton epulsion, and the final tem inludes the oetion due to nonlassial eleton eleton effets. l ψ( ) = εψ ( ) Cuent ahievements Eletoni stutue alulations povide valuable infomation about a wide ange of impotant physial and hemial popeties. One of the most impotant is the equilibium moleula geomety, the aangement of atoms that esults in the lowest enegy fo the moleule. The alulation of equilibium bond lengths and bond angles supplements expeimental data obtained fom stutual studies suh as X-ay ystallogaphy (Setion 9.3), eleton diffation (Setion 9.4), and miowave spetosopy (Setion 0.3). Futhemoe, analyses of the moleula potential enegy uve an yield vibational fequenies fo ompaison with esults fom infaed spetosopy (Setion 0.6) as well as moleula dipole moments. 6.0 Compaison of alulations and expeiments The hoie of an eletoni stutue method to solve a hemial poblem is not usually an easy task. Both the hemial auay assoiated with the method and the ost of the alulation (in tems of omputational speed and memoy) must be taken into aount. An ab initio method suh as full CI o MP, eah of whih is apable of yielding auate esults on a moleule with a small numbe of atoms and eletons, is often omputationally impatial fo many-eleton moleules. In ontast, a semiempiial o DFT alulation might make an eletoni stutue alulation on the lage moleule feasible but with an aompanying saifie in eliability. Indeed, no single methodology has been found to be appliable to all moleules. Howeve, the pomise that omputational hemisty has to enhane ou ability to pedit hemial and physial popeties of a wide ange of moleules is suffiient to dive futhe development of eletoni stutue methods. Fist onside moleula hydogen, the subjet of most of the illustations in this hapte. To ompae esults fom diffeent eletoni stutue methods, we need to say a few wods about the basis set used in the alulations. A minimal basis set uses the fewest possible basis set funtions (Setion 6.3). Howeve, the Hatee Fok limit is ahieved by the use of an infinite numbe of basis funtions. Although this limit is not omputationally attainable, a finite basis is onsideed to have eahed the limit if the enegy, equilibium geomety, and othe alulated popeties have onveged and do not vay within a speified toleane upon futhe ineases in the size of the basis set. The esults pesented fo Hatee Fok alulations that use suh a basis set ae labelled HF limit in the aompanying tables. (In patie, the HF-limit in the tables oesponds to a 6-3+G** basis; see Table 6..) So that we an ompae diffeent eletoni stutue methods moe dietly, we epot liteatue esults whee the same o a simila basis set was used in the diffeent

16 6 COMPUTATIONAL CHEMISTRY 87 Table 6. Compaison of methods fo small H-ontaining moleules Expt HF limit MNDO PM3 CI* MP DFT R(H-H)/pm R(O-H)/pm in H O Bond angle/ in H O Dipole moment, µ(h O)/D * Fo dihydogen, full CI. Fo wate, CI with inlusion of singly and doubly exited deteminants. D (debye) = C m. types of alulations. The density funtional theoy alulations to whih we efe all used the exhange oelation potential fo a unifom eleton gas, inluding the oelation omponent negleted in the fist illustation of Setion 6.9. Table 6. ompaes the equilibium bond length fo dihydogen detemined fom vaious eletoni stutue methods; the equilibium geomety oesponds to the minimum in the alulated moleula potential enegy. Not supisingly, the CI and MPPT ab initio methods ae the most auate. Howeve, fo this simple moleule the Hatee Fok esult is also within hemial auay (about pm); the semiempiial methods do not fae as well by ompaison but, as also shown in Table 6., MNDO and PM3 ae moe auate fo alulations on wate than on dihydogen. The CI and MP methods also ahieve hemial auay (within ) fo the bond angle in wate. As fo the dipole moment of wate, the semiempiial methods ae found to be slightly moe auate than the Hatee Fok and density-funtional alulations. Table 6.3 shows some esults fom semiempiial, MPPT, and DFT alulations of abon abon bond lengths in a vaiety of Table 6.3 Compaison of methods fo small ogani moleules Expt PM3 MP DFT R(C-C)/pm popane ylobutane R(C=C)/pm popene ylobutene #(C=C steth)/m popene ylobutene small ogani moleules as well as the C=C stething wavenumbes in the alkenes. As we shall see in Setion 0.7, vibational wavenumbes depend on the foe onstants fo displaements fom the equilibium geomety, and they in tun depend on the seond deivatives of the potential enegy with espet to the displaement. The methods geneally do a good job of pediting bond lengths of the single and double bonds and, even though the semiempiial methods do not pefom as well in alulating vibational wavenumbes, the esults fom Table 6.3 do give us a easonable level of onfidene in the peditive abilities of DFT and semiempiial alulations. Confidene in DFT and semiempiial methods beomes patiulaly impotant when the ost of omputations makes ab initio methods impatial; suh is the ase fo typial inogani and oganometalli ompounds. Hatee Fok methods geneally pefom pooly fo d-metal omplexes and ab initio methods an be pohibitively ostly. Howeve, DFT and semiempiial methods (suh as PM3, whih inludes paametes fo most d metals) have vastly impoved the pefomane of appliations of eletoni stutue theoy to inogani hemisty. 6. Appliations to lage moleules In the aea of themodynamis, omputational hemisty is beoming the tehnique of hoie fo estimating the enthalpies of fomation (Setion 4.8) of moleules with omplex theedimensional stutues. It also opens the way to exploing the effet of solvation on enthalpies of fomation by alulating the enthalpy of fomation in the gas phase and then inluding seveal solvent moleules aound the solute moleule. The numeial esults should uently be teated as only estimates with the pimay pupose of pediting whethe inteations with the solvent inease o deease the enthalpy of fomation. As an example, onside the amino aid glyine, whih an exist in a neutal (NH CH COOH) o zwitteioni ( + NH 3 CH CO ) fom. It has been found omputationally that, wheeas in the gas phase the neutal fom has a lowe enthalpy of fomation than the zwitteion, in wate the opposite is tue beause of stong inteations between the pola solvent and the hages in the zwitteion. Theefoe, we might suspet that the zwitteioni fom is the pedominant one in pola media, as is onfimed by potonation/depotonation alulations of the type aied out in intodutoy hemisty ouses. Computational hemisty an be used to pedit tends in eletohemial popeties, suh as edution potentials (Setion 7.6). Seveal expeimental and omputational studies of aomati hydoabons indiate that deeasing the enegy of the lowest unoupied moleula obital (LUMO) enhanes the ability of the moleule to aept an eleton into the LUMO, with an attendant inease in the value of the moleule s edution potential. The effet is also obseved in quinones and flavins, whih ae o-fatos involved in biologial eleton

17 88 6 COMPUTATIONAL CHEMISTRY tansfe eations. Fo example, stepwise substitution of the hydogen atoms in p-benzoquinone by methyl goups (-CH 3 ) esults in a systemati inease in the enegy of the LUMO and a deease in the edution potential fo fomation of the semiquinone adial (): The edution potentials of natually ouing quinones ae also modified by the pesene of diffeent substituents, a stategy that impats speifi funtions to speifi quinones. Fo example, the substituents in oenzyme Q ae lagely esponsible fo positioning its edution potential so that the moleule an funtion as an eleton shuttle between speifi poteins in the espiatoy hain (Impat I7.3). The eletoni stutue alulations desibed in this hapte povide insight into spetosopi popeties by oelating the absoption wavelengths and the enegy gap between the LUMO and the HOMO in a seies of moleules. Fo example, onside the linea polyenes shown in Table 6.4, all of whih absob in the UV egion. The table shows that, as expeted, the wavelength of the lowest-enegy eletoni tansition deeases as the HOMO LUMO enegy diffeene ineases. The smallest HOMO LUMO gap and geatest tansition wavelength is found fo otatetaene, the longest polyene in the goup. The wavelength of the tansition ineases with ineasing numbe of onjugated double bonds in linea polyenes and extapolation of the tend suggests that a suffiiently long linea polyene should absob light in the visible egion. This is indeed the ase fo β-aotene (), whih absobs light with λ 450 nm. The ability of β-aotene to absob visible light is pat of the stategy employed by plants to havest sola enegy fo use in photosynthesis (Impat I9.). IMPACT ON NANOSCIENCE I6. The stutues of nanopatiles Semionduto oxides, suh as TiO and ZnO, ae a majo aea of uent eseah beause they an at as photoatalysts, substanes that aeleate hemial eations upon absoption of light. Reations that an be enhaned by photoatalysts inlude the splitting of wate into H and O, and the deomposition of pollutants. Among the most popula photoatalyti mateials is TiO due to its low ost and atalyti effiieny. The method of pepaation of the bulk oxide has a stong influene on its atalyti popeties and expeiments that attempt to ontol the fom of its ystal lattie have been undetaken widely. Similaly, thee is widespead inteest in ontolling the stutue and photoatalyti popeties of TiO on the nanomete sale. Computational studies on small lustes of TiO patiles an povide insight into effets of size on photohemial popeties of nanomete-sized mateials, the natue of oxide substate inteations, and the gowth of lage aggegates. The most stable fom of bulk TiO at atmosphei pessue and oom tempeatue is utile (Fig. 6.8), in whih eah titanium atom is suounded by six oxygen atoms and eah O is Table 6.4 Eletoni stutue alulations and spetosopi data Polyene E/eV* λ/nm (C H 4 ) * E = E(HOMO) E(LUMO). Fig. 6.8 The utile stutue of TiO (blue sphees: Ti; ed sphees: O).

18 QMA_C06.qxd 8//08 9: Page 89 6 COMPUTATIONAL CHEMISTRY Stable geometies fo TinOn lustes, with n = 0, 3, and 5, detemined fom density funtional theoy alulations. [Fom S. Hamad et al. J. Phys. Chem. B, 005, 09, 574.] Fig. 6.9 suounded by thee Ti atoms. Eah otahedon omposed of the six O atoms aound the Ti ente shaes two edges with othe otahedons. Some expeimental studies on TiO nanopatiles suggest that the nanostutue is anatase, an elongated fom of utile in whih the otahedons shae fou edges. Othe stutual distotions appea to be possible as the patile size deeases. A eent omputational study on small TinOn lustes with n = 5 has identified the most stable stutues fo nanopatiles with sizes less than nm. To aomplish the hallenging omputational task of finding the most pobable luste stutues, density funtional theoy was used to evaluate the enegy as a funtion of geomety and speialized minimization algoithms wee used to find equilibium stutues. The alulations evealed ompat equilibium stutues with oodination numbes of the Ti atoms ineasing with patile size. These stutues wee found not to be elated to anatase. Fo TiO up to Ti5O30, the lagest nanopatile studied, the stutues with lowe enegies onsisted of a ental otahedon suounded by squae base pyamids, tigonal bipyamids, and tetaheda (Fig. 6.9). The DFT alulations evealed that stutues with a small numbe of squae base pyamids ae patiulaly stable. The stable stutues found fo the vaious luste sizes an be used in futhe omputational wok to study the effets of nanostutue on the photohemial popeties of TiO. IMPACT ON MEDICINE I6. Moleula eognition and dug design A dug is a small moleule o potein that binds to a speifi eepto site of a taget moleule, suh as a lage potein o nulei aid, and inhibits the pogess of disease. To devise 89 effiient theapies, we need to know how to haateize and optimize both the thee-dimensional stutue of the dug and the moleula inteations between the dug and its taget. The binding of a ligand, o guest, to a biopolyme, o host, is also govened by moleula inteations. Examples of biologial host guest omplexes inlude enzyme substate omplexes, antigen antibody omplexes, and dug eepto omplexes. In all these ases, a site on the guest ontains funtional goups that an inteat with omplementay funtional goups of the host. Many speifi intemoleula ontats must in geneal be made in a biologial host guest omplex and, as a esult, a guest binds only to hosts that ae hemially simila. The stit ules govening moleula eognition of a guest by a host ontol evey biologial poess, fom metabolism to immunologial esponse, and povide impotant lues fo the design of effetive dugs fo the teatment of disease. A full assessment of moleula eognition between a dug and its taget equies knowledge of the full spetum of inteations disussed in Chapte 8. But we an aleady antiipate some of the fatos that optimize the fomation of host guest omplexes. Fo example, a hydogen bond dono goup of the guest must be positioned nea a hydogen bond aepto goup of the host fo tight binding to ou. We also expet that an eleton-poo egion in a host should inteat stongly with an eleton-ih egion of a guest. Computational studies of the types desibed in this hapte an identify egions of a moleule that have high o low eleton densities. Futhemoe, gaphial epesentation of numeial esults allows fo diet visualization of moleula popeties, suh as the distibution of eleton density, theeby enhaning ou ability to pedit the natue of intemoleula ontats between host and guest. Conside a potein host with the amino aid seine in a site that binds guests. Eletoni stutue methods on the seine moleule an povide eletoni wavefuntions and eleton pobability densities at any point in the moleule. Fom the eleton pobability densities and the hages of the atomi nulei, one an ompute the eleti potential (Fundamentals F.6) at any point in the moleule (exept at the nulei themselves). The esulting eleti potential an be displayed as an eletostati potential sufae (an elpot sufae ) in whih net positive potential is shown in one olou and net negative potential is shown in anothe, with intemediate gadations of olou. Suh an elpot sufae fo seine (NHCH(CHOH)COOH) is shown in Fig. 6.0 whee net positive potential is shown in blue and net negative potential in ed. The eleton-ih egions of the amino aid ae suseptible to attak by an eletopositive speies and the eleton-poo egions to attak by an eletonegative speies. Thee ae two main stategies fo the disovey of a dug. In stutue-based design, new dugs ae developed on the basis of the known stutue of the eepto site of a known taget. Howeve, in many ases a numbe of so-alled lead ompounds ae known to have some biologial ativity but little infomation is available about the taget. To design a moleule with impoved

19 90 6 COMPUTATIONAL CHEMISTRY Fig. 6.0 An eletostati potential sufae fo the amino aid seine. Positive hage is shown in blue and negative hage in ed, with intemediate gadations of olou. The ed egions of the moleule ae eleton-ih and the blue egions ae eleton-poo. phamaologial effiay, quantitative stutue ativity elationships (QSAR) ae often established by oelating data on ativity of lead ompounds with moleula popeties, also alled moleula desiptos, whih an be detemined eithe expeimentally o omputationally. In boad tems, the fist stage of the QSAR method onsists of ompiling moleula desiptos fo a vey lage numbe of lead ompounds. Desiptos suh as mola mass, moleula dimensions and volume, and elative solubility in wate and nonpola solvents ae available fom outine expeimental poedues. Quantum mehanial desiptos detemined by alulations of the type desibed in this hapte inlude bond odes and HOMO and LUMO enegies. In the seond stage of the poess, biologial ativity is expessed as a funtion of the moleula desiptos. An example of a QSAR equation is: Ativity = 0 + d + d + 3 d + 4 d +... (6.7) whee d i is the value of the desipto and i is a oeffiient alulated by fitting the data by egession analysis. The quadati tems aount fo the fat that biologial ativity an have a maximum o minimum value at a speifi desipto value. Fo example, a moleule might not oss a biologial membane and beome available fo binding to tagets in the inteio of the ell if it is too hydophili, in whih ase it will not patition into the hydophobi laye of the ell membane (see Impat I6. fo details of membane stutue), o too hydophobi, fo then it may bind too tightly to the membane. It follows that the ativity will peak at some intemediate value of a paamete that measues the elative solubility of the dug in wate and ogani solvents. In the final stage of the QSAR poess, the ativity of a dug andidate an be estimated fom its moleula desiptos and the QSAR equation eithe by intepolation o extapolation of the data. The peditions ae moe eliable when a lage numbe of lead ompounds and moleula desiptos ae used to geneate the QSAR equation. The taditional QSAR tehnique has been efined into 3D QSAR, in whih sophistiated omputational methods ae used to gain futhe insight into the thee-dimensional featues of dug andidates that lead to tight binding to the eepto site of a taget. The poess begins by using a ompute to supeimpose thee-dimensional stutual models of lead ompounds and looking fo ommon featues, suh as similaities in shape, loation of funtional goups, and eletostati potential plots. The key assumption of the method is that ommon stutual featues ae indiative of moleula popeties that enhane binding of the dug to the eepto. The olletion of supeimposed moleules is then plaed inside a thee-dimensional gid of points. An atomi pobe, typially an sp 3 -hybidized abon atom, visits eah gid point and two enegies of inteation ae alulated: E stei, the stei enegy efleting inteations between the pobe and eletons in unhaged egions of the dug, and E ele, the eletostati enegy aising fom inteations between the pobe and a egion of the moleule aying a patial hage. The measued equilibium onstant fo binding of the Fig. 6. A 3D QSAR analysis of the binding of steoids, moleules with the abon skeleton shown, to human otiosteoid-binding globulin (CBG). The ellipses indiate aeas in the potein s binding site with positive o negative eletostati potentials and with little o muh stei owding. It follows fom the alulations that addition of lage substituents nea the left-hand side of the moleule (as it is dawn on the page) leads to poo affinity of the dug to the binding site. Also, substituents that lead to the aumulation of negative eletostati potential at eithe end of the dug ae likely to show enhaned affinity fo the binding site. [Adapted fom P. Kogsgaad-Lasen, T. Liljefos, U. Madsen (ed.), Textbook of dug design and disovey, Taylo & Fanis, London (00).]