學(xué)習(xí)資源量子化學(xué)課件2012a

1、Many-Electron Wave Function and operatorsReference:A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, Inc., New York, 1996Basic concepts, techniques, and notations of QC:The structure of the Hamiltonian, the form of the wav

2、e function, the matrix elements of the Hamiltonian between determinants, the Hartree-Fock method, etc.Outline2.1 The Electronic Problem2.2 Orbitals, Slater determinants, and basis functions2.3 Operators and matrix elements 2.4 Second quantization2.5 Spin-adapted configurationsA molecular coordinate

3、system: i,j = electrons; A, B = nuclei2.1 The Electronic Problem: the ratio of the mass of nucleus A to the mass of an electron. : the atomic number of nucleus A. is the Hamiltonian operator for a system of nuclei and electrons.In atomic units, the Hamiltonian for N electrons and M nuclei isOur main

4、 interest is to find approximate solutions to the time-independentnon-relativistic Schrdinger equation:Main features of the HamiltonianDepends on the coordinates, but not on the momenta of nucleiAs nuclei are much heavier than electrons, their velocities are much smaller. Therefore the motions of el

5、ectrons and nuclei can be described separately to a good approximation. The coupling (rovibronic states) can be taken care of later on, e.g., via perturbation theory.Adiabatic-Born-Oppenheimer ApproximationClamped (classical) nuclei:constant(1) Totally symmetric as required by “identity principle (i

6、ndistinguishability)”, meaning that any measurable quantity should not know the individual particle.(2) The nuclei are distinguishable and thus classical (geometry).(3) To specify the H, the number of electrons and external potential should be given.(internal)(4) No spin dependence although spin is

7、a NR quantity(9) No temperature dependence (zero Kelvin): using static structures to represent molecules, rather than treating them as an ensemble of molecules in a distribution of states (translational, rotational, and vibrational) corresponding to a macroscopic temperature.(5) Nonrelativistic(6) P

8、ositive definite (semi-bounded, variational principle can be applied)(7) Dealing with closed systems(8) A scalar Hermitian operatorNon-adiabatic, adiabatic & diabatic Non-adiabatic effects of nuclear motions: Nuclei hop on different nucleus-mass dependent PES Non-adiabatic, adiabatic & diabatic IfAd

9、iabatic approximation for nuclear motions: Nuclei stays on a single nucleus-mass dependent PES B.O approximation for nuclear motions: Nuclei stays on a single nucleus-mass INdependent PES Further neglecting the diagonal correction to Ui leads toAdiabatic & B.O. approximations differ in isotopic effe

10、cts ponent for two statesNon-adiabatic, adiabatic & diabatic What has been assumed so far is the following partitioning: Viz., the eigenstates of He are taken as the basis to expand the total wave function. This is called adiabatic representation of non-adiabatic effects for the nuclear motions.More

11、 generally, Viz., the non-eigenstates of He are taken as the basis to expand the total wave function. This is called diabatic representation of non-adiabatic effects for the nuclear motions.Non-adiabatic, adiabatic & diabatic ponent for two statesElectron-nucleus couplingFinite nucleus mass effect(a

12、diabatic representation; derivative coupling)( diabatic representation; potential coupling)(the more anharmonic the PES, the more enhanced NAC)Non-adiabatic, adiabatic & diabatic Non-adiabatic effects due to electron-nucleus coupling (finite nucleus mass),which usually slow down the reactionsNon-adi

13、abatic effects due to pure electronic effects, e.g., V12=spin-orbit coupling“two-state reactivity”Photochemistry, metalloenzyme,excited-state chemistry, catalysisComment on motions of nucleiNuclei are heavy enough for quantum effects to be almost negligible. They behave to a good approximation as cl

14、assical particles. Indeed, if nuclei showed significant quantum aspects, the concept of molecular structure (different configurations and conformations) would not have any meaning, the nuclei would simply tunnel through barriers and end up in the global minimum. Furthermore, it would not be possible

15、 to speak of a molecular geometry, since the Heisenberg uncertainty principle would not permit a measure of nuclear positions to an accuracy much smaller than the molecular dimension.Characteristics of PES TS: n=1, 1st order saddle point (reaction rate) PES: diatomic moleculeE0=Ee+ZPEE= E0+Evib+Erot

16、+Etrans thermal correctionsH=E+RT thermal enthalpyG=H-TS thermal free energyRReD0De(ZPE: quantum effect; Re: not directly observable)Reaction pathPES (independent of nuclear masses)Strategies for solving the electronic Schrdinger equationab initioDFTBorn-Oppenheimer approximation, NR approximationRe

17、strictions on the wave function: The antisymmetry or Pauli Exclusion PrincipleThe spin of an electron was introduced in an ad hoc mannerThe two spin functions are complete and orthonormal The wave function for an n-electron system can be written asA many-electron wave function must be antisymmetric

18、with respect to the interchange of the coordinate (both space and spin) of any two electrons. This is the so-called Pauli exclusion principle. and one spin coordinate An electron is described by three spatial coordinates Since the Hamiltonian makes no reference to spin, simply making the wave functi

19、on depend on spin does not lead anywhere. However, a satisfactory theory can be obtained if an additional requirement on the wave function is made, viz.Indistinguishable identical particlesThe permutation operator:Since the particles are indistinguishable, the two wave functionsmust correspond to th

20、e same state of the system, and they can differ at most by a multiplicative constant ! The eigenvalue equation of the permutation operator isFermions and BosonsFor particles of half-integral spins (fermions) the wave function must be antisymmetric (Pauli exclusion principle)For particles of integral

21、 spins (bosons) the wave function must be symmetric(how to derive the above from a higher level of theory ?)For electronsThus two electrons with the same spin have zero probability of being found at the same point in the 3D space. That is, there exists aPauli repulsion/blocking or “strong correlatio

22、n” between electrons of like spin. 2.2 Orbitals and Slater determinant2.2.1 Spin orbitals and spatial orbitals (building blocks)An orbital: a wave function for a single electron.A molecular (or atomic) orbital: a wave function for an electron in a molecule (or atom).A spatial orbital : a function of

23、 the position vector describing the spatial distribution of an electronIf is complete, any function can be expanded asA spin orbital: a wave function that describes both the spatial distribution and the spin of an electron.Each spatial orbital can form two spin orbitalsIf the spatial orbitals are or

24、thonormal, so are the spin orbitals. Generalized spin-orbital:2.2.2 Hartree ProductsCoulomb potential due to other electrons and the nuclei (point charge)In an exact theory, the Coulomb interaction is represented by the two-electron operator rij-1 (the Coulombs law). Here, electron-1 in experiences

25、a one-electron, local, Coulomb potentialConsider an electron in moving in some effective potentialDifferent operators for different orbitals; Different electrons in different spin orbitalsNamely, the two-electron potential rij-1 felt by electron-1 and associated with the instantaneous position of el

26、ectron-2 is replaced by a one-electron potential, obtained by averaging the interaction rij-1 of electron-1 and electron-2, over all space and spin coordinates x2 of electron-2, weighted by the probability that electron-2occupies the volume element dx2 at x2. By summing over all , one obtains the to

27、tal averaged potential acting on the electron in , arising from the N-1 electrons in the other spin orbitals. Self-consistent(variational ansatz)Hartree product of spin orbitals:Hartree ProductsThe simultaneous probability of finding electron 1 in dr1, electron 2 in dr2 is just the product of separa

28、te probabilities.Namely, the Hatree product of spin orbitals is an independent-electron (statistically uncorrelated) model. However, the electronic coordinates have been treated as distinguishable here!The one- and two-particle probability distribution functions:The one- and pair-densities of the co

29、untable but indistinguishable electrons:The electrons are uncorrelated ifHartree ProductsStrongly correlated!Fermi correlation: shell structure(every spin orbital is different, in line with the Pauli exclusion)The Hartree product violates the Fermi statistics (antisymmetry principle).The one- and tw

30、o-particle probability distribution functions:The one- and pair-densities of the countable but indistinguishable electrons:Hartree ProductsFor comparison, the Hartree product for a bosonic state, where all indistinguishable particles occupy the same orbital, is indeed uncorrelated!2.2.3 Slater Deter

31、minantsConsider a two-electron case in which the spin orbitals are occupied. Two Hartree products can be formed:Clearly, the following linear combination satisfies the requirement of the antisymmetry principleShow thatIf both electrons occupy the same spin orbital, thenA usual statement of the Pauli

32、 exclusion principle:No more than one electron can occupy a spin orbital(within the orbital approximation).(a Slater determinant, after Slater,1927) can be rewritten as a determinant:rows - electronscolumns - spin orbitalsInterchanging the coordinates of two electrons, say 1 and 2, givesFor an N-ele

33、ctron system, if N-electrons occupy N-orthonormal spin orbitals(without specifying which electron is in which orbital),its wave function can be described asA short-hand notation for a normalized Slater determinant is(the diagonal elements of the determinant)the antisymmetry property of Slater determ

34、inants is:Show that:exchangeexchangeSummary In order to build in the antisymmetry principle, the N-electron wave function should at least be a determinant, which can be constructed from N-occupied orthonormal spin orbitals. The requirement on orthonormality is no loss of generality due to the fact t

35、hat a determinant is invariant with respect to unitary transformations among the occupied orbitals.Consider a two-electron system,(1) If the two electrons have opposite spins and occupy differentspatial orbitals, e.g., The pair density averaged over spin:Which is just the Hartree product pair densit

36、y (Fermi shell structure if ) viz., two electrons of opposite spin have no Fermi exchangeifThere is a finite probability of finding two electrons withopposite spins at the same point in space, which is physicalbut the probability is overestimated.In this case, i.e., two electrons occupy the same spa

37、tial orbital,the Hartree product is correlated in spin space but uncorrelated in real space(2) If the two electron have the same spin (say ), we haveIt is the second exchange term that makes same-spin two electrons correlated.Thus the probability of finding two electrons with parallel spinsat the sa

38、me point in space is zero. In other words, there is exchange correlation between electrons of parallel spin. A Fermi hole is said to exist around an electron.In summary, within the single Slater determinantal description, the motion of electrons of the same spin is correlated (Fermi exchange),but th

39、e motion of electrons of opposite spin is not.The He atom The Hartree products (in blue) are the same: an electron in 1s, the other in 2s but the exchanges (in red) are different (singlet-triplet gap): Fermi hole (probability reduction; energy lowering) for 23S Fermi heap (probability enhancement; e

40、nergy raising) for 21SThe He atom:Uncorrelated in real space but correlated in spinFermi holealong r1=r2Fermi heapat r1=r2=0.5auPauli deformation (H2 and He2)H2(Pauli repulsion between closed-shell systems)T. Helgaker, P. Jrgensen, J. Olsen, Molecular Electronic Structure Theory (2004) Contour plot

41、of the HF wave function consists of concentric circles: independent motions of electrons of opposite spin; electrons of opposite spin can appear at the same r-pointThe HF He(1s2) atomdistorted circlesThe exact He atomCoulomb hole(normalized to -1)Coulomb correlation2.2.4 The Hartree-Fock approximati

42、on (basic ideas without going into details)The simplest antisymmetric wave function, which can be used to describe the ground state of an N-electron system, is a single Slater determinant, (according to the variation principle). The best wave function of this form is the one that gives the lowestpos

43、sible energyBy minimizingwith respect to the choice of spin orbitals, one can derive an equation, called the Hartree-Fock equation,which determines the optimal spin orbitals.The Hartree-Fock (HF) approximation constitutes the first step towards either more accurate or more approximate treatments. Si

44、ngle determinantHF equationAdditionalapproximationsAddition of moredeterminantsSemi-empiricalmethodsConvergence to exact solutionThe HF equation readswhereis an effective one-electron operator, called the Fock operator, having the form,whereis the average potential experienced by the i thelectron du

45、e to the presence of all other electrons.Sincedepends on the spin orbitals of all other electrons,the HF equation is nonlinear and must be solved iteratively. The procedure for solving the HF equation is called theself-consistent-field (SCF) method. The solution of the HF equation yields a set of or

46、thonormal HF spinorbitalswith orbital energies.The HF ground state determinant isspin orbitals have the lowest energies, called the occupied spin orbitals. , in which the NThe remaining members of the set are called virtual spin orbitals. and an infinite number of virtual spin orbitals. In principle

47、:There are an infinite number of solutions to the HF equationIn practice:The HF equation is solved by introducing a finite set ofspatial basis functions,Thus, the differential equation could be converted to a set ofalgebraic equations (Roothaan equations) and solved by standardmatrix techniques.Usin

48、g a basis set ofspatial functionsleads to a set ofspinorbitals, which consist ofoccupied spin orbitals(labeled by) andunoccupied spin orbitals(labeled by).The HF ground state:Larger and larger basis sets will keep lowering the HF energyuntil a limit is reached, called the HF limit. The HF determinan

49、t is the best determinant wave function.Illustration of HF orbital spectrumAufbau principle2.2.5 The minimal basis H2 modelIn this model, each hydrogen atom has a 1s atomic orbital (AO) andmolecular orbitals (MOs) are formed as a linear combination of atomicorbitals (LCAO). (Let:in short) - the AO c

50、entered on atom 1 at(in short) - the AO centered on atom 2 atA big step1s Slater orbital for H atom: the orbital exponent, =1.0 for H.1s Gaussian orbital: the Gaussian orbital exponent.a Slater orbital. We need a linear combination of several Gaussian orbitals to mimicThe two AOsandbut they will not

51、 be orthogonal, e.g.,can be assumed to be normalized, The linear combinations ofandwill lead to two delocalizedmolecular orbitals,and.In the present case,andare determined by symmetry,and we need not solve the HF equations. (the bonding MO, gerade symmetry)(the antibonding MO, ungerade symmetry)(LCA

52、O !) For H2, a minimal basis set includes two basis functions,and.Four spin orbitals can be formed from these two MOsand.The HF ground state in this model is:In this notation, Sometimes we indicate a spin orbital by its spatial part, using a bar or lack of a bar to denote whether it hasorspin functi

53、on. ThusThree different representations ofthe HF g.s. of minimal basis H2Molecular orbitals for H22.2.6 Excited determinantsis the best approximation to the groundstate, of the single determinant form.However, the number of N-electron determinants that could be formedfrom the 2K spin orbitals is the

54、 binomial coefficientThe HF ground state is just one of these.A convenient way of describing these other determinants is to considerthe HF ground state to be a reference state and to classify other possibledeterminants as n-excited determinants, in which n occupied spin orbitalsof the HF ground stat

55、e are replaced by virtual spin orbitals.A singly excited determinant (n=1):A doubly excited determinant (n=2):A triply excited determinant (n=3):and so on.Excited Slater determinants from a HF referenceIt should be mentioned that the excited determinants are not muchaccurate representations of the e

56、xcited states of the system. (orbital relaxation)(electron correlation)They are used as N-electron basis functions for an expansion of theexact N-electron states of the system (Compare with 1-electron basis).2.2.7 Form of exact wave function and configuration interaction methodSuppose we have a comp

57、lete set of functions. Any functionof a single variable can then be exactly expanded as-an expansion coefficient )(In an analogous way, any functionof two variables can beexpanded first as, withbeing held fixed, Sincecan be expanded in the complete setaswe haveSinceis required to be antisymmetric, i

58、.e.,we have the following restrictions:Proof:，we have From. Considering thatis an arbitrary two-variable function,one can deduce, and Using the above results, Thus, an arbitrary antisymmetric function of the two variables can be exactly expanded in terms of all unique determinants formed from a complete set of one-variable functions. In general, the above argument is readily extended to more than two variables. Configuration interaction (CI) method: from a complete set of spin orbital,The exact wav