量子化学与群论基础 6

单击此处编辑母版标题样式,*,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Quantum number,Integer values,Quantized,quantity,n,l,m,m,s,n, 1,0 ,l,n,-l,m,l, 1/2,Energy,Magnitude of orbital angular momentum,z-component of orbital angular momentum,z-component of spin angular momentum,n =,1,2,3,l,= 0,1,2,n,-1,m =,0, 1,2,l,m,s=, 1/2,The states of the single-electron atoms,6.3,Many-electron atoms,1,The,Schr,dinger,equation of many-electron atoms,（,Born-,Oppenheimer,Approximation,）,Unfortunately, precise solutions are not available through the,Schr,d,inger,equation, even for the simplest many-electron, helium, because,Independent particle model,The,Schr,dinger equation,Separation of variables,Mean field model,An electron at a distance r from the nucleus experiences a,Coulombic,repulsion from all the electrons within a sphere of radius r and which is equivalent to a point negative charge located on the nucleus.,，,n,=1,2,3,Symmetric,Bosons,Antisymmetric,Fermions,The,Pauli,principle,All electronic,wavefunctions,must be,antisymmetric,under the interchange of any two electrons.,2,Identical particles and the,Pauli,principle,Identical particles,Identical particles cannot be distinguished by means of any intrinsic properties.,Slater,determinant,Normalization constant,(i),(ii) No two electrons in an atom can have the same values for all four quantum numbers.,4,Electron configurations,The,Pauli,exclusion principle,No two electrons in an atom can have the same values for all four quantum numbers.,Ground state electron configuration ,Aufbau,principle,Hund,s rule,Electrons occupy the,orbitals,of a,subshell,singly until each orbital has one electron.,p,6, d,10, f,14,p,3, d,5, f,7,p,0, d,0, f,0,Atomic units,1,a.u mass=the mass of electron,m,= 9.10910,28,g,1,a.u charge = the charge of proton e =1.60210,-19,C,1,a.u length =Bohr radius,1,a.u energy =,e,2,/,a,0,= 27.2eV,The H,2,+,has two protons and one electron and can be described using the,Schr,dinger equation,5,Molecules,5.1,HydrogenMoleculeIon,（H,2,+,）,The,Schr,dinger equation of H,2,+,The,Schr,dinger equation of H,2,+,in a.u,The Hamiltonian,a.u,Schr,dinger equation,The variation theorem,The variation theorem for a linear expansion,The estimated wave function,The estimated wave function has to satisfy some conditions.,Note that we have to use the correct Hamiltonian for the system, but we do not know how to solve the,Schr,dinger equation for this Hamiltonian. The variation theorem tells us that:,E,The 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 energy,E,The problem is a maximum-minimum problem in calculus. We must have:,The wave function,The solution of,Schrodinger,equation of H,2,+,LCAO-MO,R,r,a,The estimated wave function,If,R,r,a, then,The energy of H,2,+,All the integrals above can in principle be evaluated. We know the functions and the operator. We will just give them names:,so,These equations are called,linear homogeneous equations,.,The secular determinant,H,aa,=,H,bb,H,ab,=,H,ba,S,ab,=,S,ba, and,c,1,=,c,2,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 of that satisfy the determinant equation:,c,1,=-,c,2,Approximate,wavefunction,solve the equation for,E,1,Normalization,solve the equation for,E,2,So,Normalization,The integrals,S,ab,H,aa,and,H,ab,（,i）,S,ab,the overlap integral,R,0, so,S,ab,0.,If,R,= 0,S,ab,=1;,R,=,S,ab,= 0.,（,ii）,H,aa,Coulomb integral,(,iii）,H,ab,exchange integral(,integral),R, 0，so,H,ab,0,H,ab,R,，,H,ab,，,S,ab,1，,E,1,=,H,aa,+,H,ab,=,+,，,E,2,=,H,aa,-,H,ab,=,-,H,aa,E,a,，so,E,1,=,E,a,+,，,E,2,=,E,a,-,Discussion,（i）The energy of,1,and,2,The calculated and experimental molecular potential energy curves for a hydrogen molecule-ion.,（ii）Bonding orbital,1,The electron density calculated by forming the square of the,wavefunction,. Note the accumulation of electron density in the,internuclear,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,2,A partial explanation of the origin of bonding and,antibonding,effects. (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 Hamiltonian,A molecule consists of number of electrons and nuclei. The molecular Hamiltonian operator,has a complicated form.,=,(1,2,N,)：,(,Within the Born-,Oppenheimer,approximation),Main approximation of,ab initio,MO theory,the Born-,Oppenheimer,approximation,The orbital approximation,Non- relativity approximation,2.,The molecular,wavefunctions,(molecular,orbitals,),So lets consider a simpler problem, involving the one-electron,hamiltonian,Separation of variables,(1,2,3,N,),(1,2,N,)=,det,(1)(1) (2)(2) (,N,)(,N,) ,3.,Variational,parameter,or,D,=,C,C,D,is called the density matrix, a product of AO -MO coefficient matrices,4.,Hartree,-,Fock,equations,Lets 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 product, and is in fact an,eigenvalue,equation in the form,H c,=,S c,E,we 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,.,es,DocsKnowledgeFundamental_Theory,hf,hf,.html,Capabilities of,ab initio,quantum chemistry,Can calculate,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 the 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,5.3,The,Huckel Moleculor,Orbital method (HMO),HMO deal with conjugated molecules.,Butadiene , e.g.： 6,1s+4,(1s,2,2s,2,2p,x,1,2p,y,1,2p,z,0,)=26 AO,HMO approximation : 4,p,z,.,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.,Huckel,approximation I,HMO is suggested by,Eric H,ckel,in 1931.,Butadiene,4,p,z,of C atoms,The energy and coefficients satisfy the following equations:,let,The best molecular,orbitals,are those which,minimise,the total energy. This is achieved by imposing the condition:：,Huckel,approximation II：,non-trivial solutions ：,These values, called the,non-trivial solutions,to these equations, occur when:,let,This determinant can be easily multiplied out to give:,x,4,-3,x,2,+1=0,1,=0.3717,1,+0.6015,2,+0.6015,3,+0.3717,4,2,=0.6015,1,+0.3717,2,0.3717,3,0.6015,4,3,=0.6015,1,0.3717,2,0.3717,3,+0.6015,4,4,=0.3717,1,0.6015,2,+0.6015,3,0.3717,4,0,so,E,1,E,2,E,3,E,4,We obtain four values of,E, which is reasonable since we expect to find four molecular,orbitals,.,Delocalization,energy,Total energy,E,=2,E,1,+2,E,2,=2,(,+1.62,)+,2,(,+0.62,),=4,+4.48,Energy levels,Occupied orbital,Unfilled orbital,C = C,C = C,E,=,4,+4,E- E,=0,.48,Frontier,orbitals,The highest occupied molecular orbital, HOMO,The lowest unfilled molecular orbital, LUMO,The frontier,orbitals,are important because they are largely responsible for many of the chemical and spectroscopic properties of the molecule.,