學(xué)習(xí)資源量子化學(xué)課件2012b

1、The Hartree-Fock methodThe SCF procedure, illustrations, basis sets, etcSpecify a molecule ( ), a basis set and the STATE3.3.6 The SCF procedure 3.3 Restricted Closed-Shell Hartree-Fock: the Roothaan EquationsThe SCF procedure electronAANZR,Calculate all required molecular integrals, , , and Diagona

2、lize the overlap matrix S and obtain a transformation matrix XObtain a guess at the density matrix PCalculate the matrix G from the density matrix and the two-electron integralsuvScoreuvHuv1)2)3)4)5)Diagonalize to obtain and Calculate Add G to the core-Hamiltonian to obtain the Fock matrix Calculate

3、 the transformed Fock matrix Form a new density matrix P from C using Determine whether the procedure has converged. If not, return to step 5 with the new density matrix GHFcoreFXXFFCXCC 22NavauauvCCPIf the procedure has converged, then calculate expectation values and other quantities of interest6)

4、7)8)9)10)11)12)About the convergenceThe above procedure likely oscillates without interventionAbout the convergence(1) Damping P1(1)iiiPPP(2) Level-shifting F00OOOVOOVOVVVVFFFFFF(3) DIIS (Direct Inversion in Iterative Subspace),0FPS SPF FPSSPFExtrapolated F by solving a small set of linear system of

5、 equationsgradientUseful concepts: 0MAMABABBAAAtotRZZRERE10 AtotRE AR 0totAAERRMA, 2, 1 .10 ) 1 (6100uaEEii(a) The total energy and the electronic energy(b) The potential energy surface for nuclear motion: the total energyas a function of a set of nuclear coordinates(c) The equilibrium geometry:,(d)

6、 Ab initio: no approximations are made to the integrals and the electronicHamiltonian.(e) The SCF convergence criterion:. 42121210matrixdensity )2(uviuviuvPPK.3.3.7 Expectation values and population analysis ar: a poor approximation to the observed electron affinities. (1) Orbital energies: a reason

7、able approximation to the observedionization potentials. uvuvcoreuvvuNbababaaaaaNaaaaNaNaNbababaaFPKJhfhKJhHE21 2 22 22222000 AABABBAtotRZZEE0(2) The electronic and total energies Niih11 uvuvNaaauhvPh201012 nucleiAAAelectronNiiiiimomentdipoletheRZrrq01000uvAAAuvRZurvPAAAuvuvxXZuxvP(3) One-electron e

8、xpectation values(not necessarily the core-Hamiltonian)For example,This is a vector equation with components PStrPSSPrrdrrdrrdNuuuuvvuuvNaaNaa 222222(4) Population analysis AuuuAAPSZq(a) Mulliken population:The net atomic charge associated with atom A is given byAuuuPSis the gross atomic population

9、,ANAZbasis functions centered on atom A. whereis the charge ofatomic nucleus A. The index of summation indicates that we only sum over the uuuuuuPSSPSN2121AuuuAAPSSZq2121(b) Lwdin population:AOIM: Liu and Li, Theor. Chim. Acta 95, 81 (1997)uuuuuuSPPSNu uvvurrrdIS uuuPNvvvuuX22NavauauvCCPandXCC yield

10、 Proof:Since it is always possible to find a transformed set of functionsSupplementary Materialsuch that, i.e., thus with this set ofnew basis functionsUsing uvkllvklukXvlNalakaklukNalalvlkakukNavauauvXXPXPXXCCXCXCXXCXCPlv222222XXPP11XPXPuuuXPXN1121 SX1SXXuuuuuuPSSSSSSSSSPSN21212121121121 todue , No

11、te which completes the proof. ThusHenceSince the matrixsatisfies, we haveComparison of a Slater function with a Gaussian function:(a) Least squares fit of a 1s STF (1.0, exact H 1s orbital) by a single 1s GTF (0.270950)(b) Comparison of the corresponding radial distributionNote that the different be

12、haviors at small and large r !3.4 Model Calculations on H2 and HeH+3.4.1 The 1s Minimal STO-3G Basis setGTOs lead to efficient analytical integrals but do not have the desired functional behavior. One way around is to use contracted GTO to fit a STO (STO-nG) .The product of 2 GTOs is another GTO2|3/

13、412( ,)()Ar RGFsArRe3/422()exp| ()()ABABKRR 111( ,)( ,)( ,)GFGFGFsAsBABsPrRrRKp rRpABPRRR0ABR112 111212(|)( ,)( ,)GFGFABCDABCDsPsQKKdrdrp rRrq rR ABCDPQKKR1,ACRifAB CD()0AB CDforR GTO vs. STOThe -dependence makes GTO inferior to STO At the nucleus GTO has zero slope, in contrast to STO which has a “

14、cusp” (discontinuous derivative). Therefore, GTOs have problems representing the proper behavior near the nucleus.GTO falls off too rapidly far from the nucleus compared with STO, and the “tail” of the wavefunction is consequently represented poorly.However, the product of 2 GTOs is still a GTO ! su

15、ch that 4-centered 2e integrals can be reduced to 2-centered ones.2r3.4.2 STO-3G H2121.00ABSatR120ABSifR 1.00.65930.65931.0S0.76000.23650.23650.7600T1.22660.59740.59740.6538AV0.65380.59740.59741.2266BV1.4ABRau11()0.444635(0.168856,)GFAsArRR110.535328(0.62391,)0.154329(3.42525,)GFGFsAsARR3|1/21.(2)4(

16、)Ar Re11220.5TT11221ABVV 1ifA larger exponent ( ) leads to a smaller orbital than in free H ( ), reflectingthe fact that the H2 is smaller than the sum of 2H. The electron is therefore closer to the nucleus, leading to a more negative potential (-1.2266 vs. -1), and it “travels faster to avoid colla

17、psing into the nucleus”, leading to a larger kinetic energy (0.7600 vs. 0.5). The energy of a H atom in this basis is just 1.241.011110.7600 1.22660.4666ATVau to be compared with the exact value of -0.5 au. If an electron in were localizedexactly at the position of nucleus A, its attraction for nucl

18、eus B would be 11/1.40.7143 The actual value of this attraction is, at R=1.4 au, 110.6538BVau 1110ABRBABVR As the internuclear distance increases,The off-diagonal elements of T and V cannot be given such simple classical interpretations, and they constitute the basic quantum effects of bonding. As t

19、he internuclear distance es large, the off-diagonal elements go to zero, i.e., no bonding exists anymore.The core-Hamiltonian matrix is (full for H2+) 1.12040.95840.95841.1204coreABHTVVThere are 4 unique 2e integrals among 24=16(11|11)(22|22)0.7746(11|22)0.5697(21|11)(21|22)0.4441(21|21)0.2970auauau

20、auThe one-center integrals (11|11) and (22|22) represent the average value of the electron-electron repulsion of two electrons in the same 1s orbital.The two-center integral (11|22) is the repulsion between an electron in anorbital on center A and an electron in an orbital on center B. Its value, wh

21、ich is0.5697 au at R=1.4 au, will tend to 1/RAB as the internuclear distance increases.There are no classical interpretations on the integrals (21|11) and (21|21). They both go to zero at large RAB as the overlap S12 goes to zero.Having prepared all the quantities, we are now ready to solve the HFR

22、equations.However, we here have only two orbitals, which are determined by the orthogonalitycondition and symmetry, i.e., we do not need an iterative procedure.1/2121/211212121/2122(1)2(1)()( ,)2(1)gSSS 1/2121/211212121/2122(1)2(1)()( ,)2(1)uSSS In this MO basis, the relevant integrals are1111|1.252

23、8hh 2222|0.4756hh 111111(|)0.6746J 222222(|)0.6975J 121122(|)0.6636J 121221(|)0.1813K /21|(2)NiiiiiiaiaafhJK111110.5782hJ 222212120.6703hJK Note that 22222,hJsince it is a virtual orbital and describes the energy of an electron in the (N+1)-electron system (not self-interaction free)0111121.8310EhJ

24、The electronic energy for the ground state is The total energy, including nuclear repulsion, is 2011.1167HABEER The dissociation energy is222*( 0.4666) 1.11670.18354.99eHHDEEaueV(experimental value: 4.75 eV)(The error in the molecule is cancelled by that in the atom)Molecular orbitals for H2The RHF

25、PES of STO-3G H2RHF does not dissociate H2 to two H atoms !/2*2(|)(|)NcoreaaaFHC C 1(|)(|)2coreHP /201()()2NcoreaaaaaEhfPHFSTO-3G H211122111211(1) (11|11)(11|22)(11|12)(12|12)22coreFFHS11221121213(1) (11|22)(11|12)(12|12)22coreFFHS112(1)PSH-1When R 110BV(11|22)0(11|12)0(12|12)0120S111111()coreAHTVE

26、H111(11|1( )21)2coretotERH2*( 0.4666)0.5*0.77460.54592 ()E H 120F InterpretationsSince both electrons occupy the same spatial orbital, the spurious repulsion remains finite even at infinity. The products of the dissociation are not just 2H, but also include, incorrectly, H- and H+.Alternatively, thi

27、s can be understood as following: a molecular orbital wave function is equivalent to a valence bond wave function in which equal weight is given to covalent and ionic terms. Due to symmetry reasons, the ionic term survives even on dissociation.(11|11)(It is generally true that RHF gives the wrong li

28、mit, if the products are open-shells)An SCF Calculation on STO-3G HeH+11()()HeHXHe SH Because the IP of He (24.6 eV) is larger than the EA of a proton (13.6 eV), thedissociation products are He and H+, instead of He+ and H12| 1.4632RRRauWe will have to solve the HF iteratively !The initial guess for

29、 P is, e.g., to diagonalize the core Hamiltonian matrix2.65271.34721.34721.7318coreFH2.67410.000000.000001.30430.92910.62590.13981.1115Cto obtainwhich are the orbitals and orbital energies of HeH+Note that the lower MO is composed mainly of He 1s (0.9291), because in the absence of e-e repulsion the

30、 electrons tend to concentrate near the larger nucleus(He). The effect of adding e-e repulsion will be to moderate this effect and “smear”the electrons out a bit, so as to decrease the e-e repulsion.From the MO coefficients we can form our first real guess at P (representing twonon-interacting elect

31、rons in the field of the nuclei) 1.72660.25990.25990.0391PThe diagonal elements of P show, only qualitatively, how most of the electron densityis in the vicinity of the He rather than the H nucleus. From P we can construct G:1.26230.37400.37400.9890Gand a new Fock matrix F1.39040.97320.97320.7429cor

32、eFHGBecause of the positive e-e interaction, represented by the positive elements of G,the elements of this new F are considerably less negative than the original coreHamiltonian guess. We can now solve the eigenvalue problem with this latest F to get a new guess at C and P. Repeat the whole procedu

33、re until self-consistency is achieved.To provide a variational value of the energy at each iteration, the formula01()2coreEPHFmust use the same P as was used to form F. Thus the energy should be calculatedimmediately after forming a new F, not immediately after forming a new P. IterationP11P12P22E0(

34、au)11.72660.25990.0391-4.14186321.33420.51660.2000-4.22649231.28990.53840.2247-4.22752341.28640.54000.2267-4.22752951.28620.54020.2269-4.22752961.28610.54020.2269-4.227529STO-3G HeH+: Density matrix and energy during iterationsBecause the energy is made variational, the relative error in the energy

35、is less thanby one order of magnitude in the wave function or density matrix. Namely, the energyconverges faster than the density matrix.The final wave function and orbital energies are 0.80190.78230.33681.0684C1.59750.000000.000000.0617Envisaged by the signs for the two coefficients, the lower orbi

36、tal is a bonding orbital.It is still composed mainly of He 1s. The virtual orbital is an antibonding orbital withopposite signs for the coefficients. It has a larger coefficient for H 1s due toorthogonality.Koopmans theorem: IP=1.5975 au=43.5 eV EA=0.0617 au= 1.7 eVThe positive EA predicts that HeH+

37、 will bind an electron. However, this does not mean that HeH be a stable molecule, since the dissociation products He+H+ bindan electron much more strongly (the EA of H+ is greater than the sum of the EA and dissociation energy of HeH+) .Mulliken population analysis: He+0.47H+0.53Lwdin population an

38、alysis: He+0.53H+0.47Dissociation channel:0.2168HeHHeHEau0.4168HeHHeHEauThe RHF PES of STO-3G HeH+101111()22(11|11)()E RTVE He 10()01C R 20()00P R RHF dissociates correctly if the products are closed-shells3.5 Choice of basis setsAn art !An art !IntroductionlThe Holy Grail of computational chemistry

39、 is the calculation of the molecular orbitals (MOs) for a given molecule. From the MOs we can know lots of things about the molecule such as: lenergy lelectron density lelectrostatic potential ltransition state (if any) lfrequency However, calculating MOs is NOT that easy. Nonetheless, the computer

40、does this for us.Chemists should first tell the computer some information via Input files. Geometry (Cartesian or internal coordinates)Bond lengths Bond Angles Dihedrals Kind of Calculations: Single point energy Frequency Transition state Electron density Electrostatic potential Starting set of math

41、ematics and approximations Calculation method (semiempirical, or ab initio type of approximation (Hartree-Fock, Moller-Plesset, etc.) )Basis Set Approximation What Chemists and Computers Should DoThe graphic below captures the essence of what is the responsibility of the chemist and what is the resp

42、onsibility of the computer: What we will concentrate in this chapter Basis set expansion of MO( )( )iirCrPhilosophy lIn principle, any complete functions can be usedlThe chosen functions should be able to describe the desired properties l (cusps; different environments: polarization, diffusion, vari

43、ous states).lEfficient evaluation of the integralsFunctions of different formslHydrogenic:lSlater:lGaussian:lNumerical: grid-based (single-site orbital)lPlane wave:( )exp)(AnlZfrrn, , ,1,( , , )exp()n l mAnl mrNYrr exp()ik r, , ,(22),2( , , )exp()n l mnll mArNYrr Contracted basis sets(contraction le

44、ngth, coefficients, exponents)lSegmented contractionlGeneral contraction611()()iiiCGTOaPGTO927()()iiiCGTOaPGTO310()()CGTOPGTO101()()jijiiCGTOcPGTO 2413318,rserg241352128,rpxergx2413732048,rdxyergxy6.6 Polyatomic Basis Sets KuuuiiC1LpppuppuACGFuRrgdRr1,ppupRrg,pth normalizedpuorbital exponent, center

45、ed at .pR: ThepudL: The contraction coefficient.: The length of the contraction. primitive Gaussian with the Gaussian LpRp, 2, 1ARare almost chosen to be2( ,)() () () exp(| )mnlAAAAAgRN xXyYzZrRSCF calculations. The orbital exponents, which are positive numbers, determine thediffuseness or “size” of

46、 the basis functions (a small exponent impliesa large diffuse function). The contracted basis functions might be chosento approximate Slater functions, Hartree-Fock atomic orbitals, etc. ForFor hydrogen (from SCF calculations on H) rgrgrgrgsssss,3615.1301906. 0,01330. 213424. 0,453757. 047449. 0,123

47、317. 050907. 011111s4 The basis set is a basis. example, Huzinaga determines contractions from the results of atomic into one basis function, i.e., rgrs,123317. 011 rgrgrgNrsss,3615.1301906. 0,01330. 213424. 0,453757. 047449. 01112 ss24This scheme defines asegmented contraction.A contracted basis se

48、t derived from this would use the four Gaussianfunctions as primitives and contract them to reduce the number of basisfunctions. A common contraction scheme is one that leaves the mostdiffuse primitive uncontracted and contracts the remaining three primitivesps59psps2359sps4/59sps2/23For the first-r

49、ow atoms Li to Ne, Huzinaga determined uncontractedGaussian basis sets. Dunning suggested useful contractions of these.For O,H2O, the uncontracted basis set; the contracted basis might be Classification of basis sets (exponents and size N)There are two general categories of basis sets:Minimal Basis

50、Sets (STO-nG)The smallest number of functions (describing only the most basic aspects of the MOs), e.g., 1s for H and He, 2s and 1 set of p for Li-NeExtended Basis Sets A basis set that describes the MOs in great detail, e.g., DZ, TZ, QZ, 5ZTZ2P,exp()HFEL3(1)CELPople Style Basis sets: k-nlmG(segment

51、ed contraction)lk PGTOs for core orbitals lnlm indicate both how many functions the valence orbitals are split into, l and how many PGTOs are used for their representation. lTwo values (nl) means a split valence DZ, three values (nlm) means l a split valence TZ. l The values before G indicate sp-fun

52、ctions, whereas after G are the l polarization functions.lThe same exponents are used for sp-functions l (better efficiency, worse flexibility).lProblem: no systematic convergence !The Pople Style Basis sets: C/Hs=k+n+l+m p=n+l+mlBackbone:l 3-21G: ( 6s3p/3s)3s2p/2sl 6-31G: (10s4p/4s)3s2p/2sl 6-311G:

53、 (11s5p/5s)4s3p/3slDiffuse sp (“+”) and polarization functionsl 6-31+G(d): diffuse sp and polarization d for non-Hl 6-311+G(2df,2pd): diffuse sp; 2d, 1f for non-H l diffuse s ; 2p, 1d for HlAn asterisk means only 1 set of polarization functionl 6-31G*=6-31G(d), 6-31G*=6-31G(d,p)STO-2G for H and C BA

54、SIS=STO-2G H 0 S 2 1.00 1.30975638 0.43012850 0.23313597 0.67891353 * C 0 S 2 1.00 27.38503303 0.43012850 4.87452205 0.67891353 SP 2 1.00 1.13674819 0.04947177 0.51154071 0.28830936 0.96378241 0.61281990contraction coefficient: s-part of the sp-hybridcontraction coefficient: p-part of the sp-hybride

55、xponent 6-31G*=6-31G(d) for H & CBASIS=6-31G* H 0 S 3 1.00 18.73113700 0.03349460 2.82539370 0.23472695 0.64012170 0.81375733 S 1 1.00 0.16127780 1.00000000 * C 0 S 6 1.00 3047.52490000 0.00183470 457.36951000 0.01403730 103.94869000 0.06884260 29.21015500 0.23218440 9.28666300 0.46794130 3.1639

56、2700 0.36231200 SP 3 1.00 7.86827240 -0.11933240 0.06899910 1.88128850 -0.16085420 0.31642400 0.54424930 1.14345640 0.74430830 SP 1 1.00 0.16871440 1.00000000 1.00000000 D 1 1.00 0.80000000 1.000000006 PGTOs for core orbital3 PGTOs for one valence orbital1 PGTO for the other valence orbital1 d PGTO

57、for polarization S 3 1.000.443228850E+04 0.630928000E-010.670660120E+03 0.374503800E+000.146902450E+03 0.683416000E+00 SP 3 1.000.195004187E+03 -.111628337E+00 0.143805463E+000.425688915E+02 0.943355284E-01 0.570001877E+000.131214345E+02 0.961100247E+00 0.453311855E+00 SP 3 1.000.134023055E+02 -.291

58、781114E+00 0.287052823E-010.439990638E+01 0.342614548E+00 0.486251467E+000.138514757E+01 0.848283984E+00 0.590235252E+00 D 2 1.000.183682020E+02 0.275385631E+000.459130410E+01 0.843477290E+00 D 1 1.000.109020260E+01 0.100000000E+01 SP 2 1.000.112155801E+01 -.202370626E+00 0.344094080E-030.122943640E

59、+00 0.107703478E+01 0.999905284E+00 SP 1 1.000.421932724E-01 0.100000000E+01 0.100000000E+012s2pCore:3 PGTOs 1 CGTO Valence:3 PGTOs 2 CGTOs3-21G for 30Zn Ar3d104s2 3d4s4p(12s9p3d)5s4p2dDunning correlation consistent (cc) basis sets (general contraction)l“cc” refers to the fact that functions which c

60、ontribute similar amounts of correlation energy are included at the same stage, independently of the function type.lPolarization pattern: 1d, 2d1f, 3d2f1g, 4d3f2g1h, 5d4f3g2h1ilcc-pVXZ, aug-cc-pVXZ, cc-pCVXZlAdvantage: systematic convergencelDisadvantage: too large(3,4,5,6)LEABeL312,NL tLLinear dependen