1、6.3 Many-electron atoms 1 The Schrdinger equation of many-electron atoms（Born-Oppenheimer Approximation）2ji2ji2jiijzzyyxxr Unfortunately,precise solutions are not available through the Schrdinger equation,even for the simplest many-electron,helium,becauseNjiijNiiNiirerZem212122212HEH Independent par
2、ticle model021jiijNerThe Schrdinger equationErZemNiiNii121222Separation of variables Nn3213,2,1NjiijNiiNiirerZem2121222HiiirZem2222HNiitotalEEEEEE1n321-NNH.22H11HNNNN22221111EEE),()(),(,mllnmlnYrRrRnZEn22 Mean field model iiiiii22i22ErVrZeAn electron at a distance r from the nucleus experiences a Co
3、ulombic repulsion from all the electrons within a sphere of radius r and which is equivalent to a point negative charge located on the nucleus.i2iirerUiiii2i22ErZRnZEn22，n=1,2,3,221,22,11,22,11,22,1Symmetric,Bosons 1,22,1Antisymmetric,Fermions The Pauli principle All electronic wavefunctions must be
4、 antisymmetric under the interchange of any two electrons.2 Identical particles and the Pauli principle Identical particlesIdentical particles cannot be distinguished by means of any intrinsic properties.Slater determinant NNNNNNNNNNNNNN221122112211!1,2,1222222111111）（!1N Normalization constant(i)1,
5、22,1(ii)No two electrons in an atom can have the same values for all four quantum numbers.4 Electron configurations The Pauli exclusion principleNo two electrons in an atom can have the same values for all four quantum numbers.Ground state electron configuration Aufbau principle Hunds ruleElectrons
6、occupy the orbitals of a subshell singly until each orbital has one electron.p6,d10,f14p3,d5,f7p0,d0,f0 Atomic units 1a.u mass=the mass of electron m=9.1091028g 1a.u charge=the charge of proton e=1.60210-19C 1a.u length=Bohr radiusm100.52910-220mea 1a.u energy =e2/a0 =27.2eVRererem2b2a2222HE HThe H2
7、+has two protons and one electron and can be described using the Schrdinger equation 5 Molecules 5.1 Hydrogen Molecule Ion（H2+）The Schrdinger equation of H2+The Schrdinger equation of H2+in a.u The Hamiltonian Rererem2b2a2222Ha.uRrr11121Hba2 Schrdinger equationERrr11121ba2 The variation theorem The
8、variation theorem for a linear expansiond*dH*E The estimated wave function The estimated wave function has to satisfy some conditions.iiccccNN2211Note that we have to use the correct Hamiltonian for the system,but we do not know how to solve the Schrdinger equation for this Hamiltonian.The variation
9、 theorem tells us that:EThe expectation value of the energy is always higher than the correct result.Molecular Orbital-a Linear Combination of Atomic Orbitals LCAO-MO Expectation value of the energyENiccccfE,21The problem is a maximum-minimum problem in calculus.We must have:021NicEcEcEcE The wave f
10、unction The solution of Schrodinger equation of H2+ERrr11121ba2barrececcc1121b2a1 LCAO-MOR,ra,b1erba2121Hrararsaeea11001 The estimated wave function If R,ra,thena1are The energy of H2+d*dH*Ebarrececcc1121b2a1 ddH2b2a1b2a1b2a1ccccccEdddddHdHdHdH2b22ab21ba212a21bb22ab21ba21aa21ccccccccccccAll the inte
11、grals above can in principle be evaluated.We know the functions and the operator.We will just give them names:dHaaaaHdHbaabHdHabbaHdHbbbbHdbaabSdabbaS1dd2b2abbaaSSso22ba21ab2121bb22ba21ab21aa21cSccScccHcHccHccHcE,0,021cEcE002bb2baba2abab1aacEHcESHcESHcEHThese equations are called linear homogeneous
12、equations.0bbbabaababaaEHESHESHEHThe secular determinantHaa=Hbb,Hab=Hba,Sab=Sba,and02abab2aaESHEHababaa11SHHE c1=c2 The important question is whether there is a solution other than the trivial solution.There is.The wave function disappears(the trivial solution)for all values of except for the values
13、 of that satisfy the determinant equation:ababaa21SHHE c1=-c2 Approximate wavefunctionsolve the equation for E1ba11 cNormalizationbaab1221Ssolve the equation for E2ba12 cSoababaa11SHHEbaab1221Sababaa21SHHEbaab2221Sbaab2221SNormalization The integrals Sab,Haaand Hab（i）Sabthe overlap integral dbaabSRe
14、RRS131d2baabbr-be1ar-ae1R 0,so Sab 0.If R=0,Sab=1;R =,Sab=0.RabeRRdRdS231（ii）HaaCoulomb integral dHaaaaHar-ae1aaRaaaEJEeREH211(iii）Habexchange integral(integral)dHbaabHRabeRRRH12167612R 0，so Hab 0,HabR，Hab，ar-ae1br-be1Sab1，E1=Haa+Hab=+，E2=Haa-Hab=-HaaEa，so E1=Ea+，E2=Ea-ababaa11SHHEababaa21SHHE Discu
15、ssion（i）The energy of 1 and 2The calculated and experimental molecular potential energy curves for a hydrogen molecule-ion.（ii）Bonding orbital 1ba1221Sba2b2aab212221SThe electron density calculated by forming the square of the wavefunction.Note the accumulation of electron density in the internuclea
16、r region.The boundary surface of a(orbital encloses the region where the electrons that occupy the orbital are most likely to be found.Note that the orbital has cylindrical symmetry.（iii）Antibonding orbital 2ba2b2aab222221S2122SabA partial explanation of the origin of bonding and antibonding effects
17、.(a)In a bonding orbital,the nuclei are attracted to the accumulation of electron density in the internuclear region.(b)In an antibonding orbital,the nuclei are attracted to an accumulation of electron density outside the internuclear region.5.2.Molecular orbital theory(MO theory)1.The molecular Ham
18、iltonianA molecule consists of number of electrons and nuclei.The molecular Hamiltonian operator has a complicated form.=(1,2,N)：EHPbaabbaNjiijPaiaiaNiiRZZrrZ2112121H2(Within the Born-Oppenheimer approximation)Main approximation of ab initio MO theory the Born-Oppenheimer approximationThe orbital ap
19、proximationNon-relativity approximation 2.The molecular wavefunctions(molecular orbitals)So lets consider a simpler problem,involving the one-electron hamiltonian NN213,2,1Separation of variablesiiiihapeaiVVrh1212Ekji.|H|ihH (1,2,3N)EH NNNNNNNNNNNNNN221122112211!1),2,1(222222111111(1,2,N)=det(1)(1)(
20、2)(2)(N)(N)3.Variational parameterNiiiic|iiCC*D,or D=CC D is called the density matrix,a product of AO-MO coefficient matrices4.Hartree-Fock equationsLets look at a general example of functional variation Writing the energy as we want E=0,so Thus It is clear that this can be written as a matrix prod
21、uct,and is in fact an eigenvalue equation in the form H c=S c Ewe can rewrite the Hartree-Fock equations as Using the fact that is diagonal,this can be written as the matrix product F C=S C www.adi.uam.esDocsKnowledgeFundamental_Theoryhfhf.htmlCapabilities of ab initio quantum chemistryCan calculate
22、 wavefunctions and detailed descriptions of molecular orbitals Can calculate atomic charges,dipole moments,multipole moments,polarisabilities,etc.Can calculate vibrational frequencies,IR and Raman intensities,NMR chemical shifts Can calculate ionisation energies and electron affinities Can include t
23、he electrostatic effects on solvation Can calculate the geometries and energies of equilibrium structures,transition structures,intermediates,and neutral and charged species Can calculate ground and excited states Can handle any electron configuration Can handle any element Can optimise geometries 5
24、.3 The Huckel Moleculor Orbital method(HMO)HMO deal with conjugated molecules.Butadiene,e.g.：61s+4(1s22s22px12py12pz0)=26 AOHMO approximation:4 pz.In his approach,The orbitals are treated separately from the orbitals,and the latter form a rigid framework that determine the shape of the molecule.Huck
25、el approximation IHMO is suggested by Eric Hckel in 1931.Butadiene44332211cccc414 pz of C atomsijijjiijijjiccccEddHddH2EHV2i22H The energy and coefficients satisfy the following equations:000044444343432424214141434343333323232131314242432323222221212141414313132121211111cESHcESHcESHcESHcESHcESHcESH
26、cESHcESHcESHcESHcESHcESHcESHcESHcESHdjiijSdHjiijHlet The best molecular orbitals are those which minimise the total energy.This is achieved by imposing the condition:：01cE02cE03cE04cE Huckel approximation II：neighbours-nonj,i,0neighbours j,i,=j=i ,=ijHji0,ji,1ijSEEEE0000000 non-trivial solutions：044
27、44434342424141343433333232313124242323222221211414131312121111ESHESHESHESHESHESHESHESHESHESHESHESHESHESHESHESH These values,called the non-trivial solutions to these equations,occur when:xEletxE0100110011001xxxxThis determinant can be easily multiplied out to give:x4-3 x2+1=0618.1618.0618.0618.10114
28、32122xxxxxxxx1=0.37171+0.60152+0.60153+0.371742=0.60151+0.371720.371730.601543=0.601510.371720.37173+0.601544=0.371710.60152+0.601530.3717462.162.062.062.14321EEEE 0,so E1 E2 E3 E4We obtain four values of E,which is reasonable since we expect to find four molecular orbitals.Delocalization energyTota
29、l energy E=2E1+2E2=2(+1.62)+2(+0.62)=4+4.48Energy levelsOccupied orbitalUnfilled orbitalC=C C=C0100100001001xxxxE=4+4E-E=0.48Frontier orbitalsThe highest occupied molecular orbital,HOMOThe lowest unfilled molecular orbital,LUMOThe frontier orbitals are important because they are largely responsible for many of the chemical and spectroscopic properties of the molecule.
