第一讲交叉点的计算

1、H T VHtot(r,R) TNTeVeeVeNVNN12M212mei2i1rijjiiZriiZZRTN He(r;R)Nuc-nuc repulsionel-nuc attractionel-el repulsionNuc kinetic EnElectr. kinetic En(TN EIe)I ETIT(r,R) I(R)Ie(r;R)HT TN He12M2 He(r;R)HeIe EIeIeHTT ETT are facilitated by the close proximity of potential energy surfaces. When the potential

2、 energy surfaces approach each other the BO approximation breaks down. The rate for nonadiabatic transitions depends on the energy gap.T(r ,R ) I(R ) Ie(r ;R )I 1NaHeIe EIeIe(TN1KII EIe)I12(2fIJJ KIJJINJ) ETIfIJ(R) IeJerkIJ(R) Ie2JerWhen electronic states approach each other, more than one of them s

3、hould be included in the expansionfIJ I JIH JEJ EIfIJ fJIfII 0I2J fIJ fIJfIJTwo adiabatic potential energy surfaces cross. The interstate coupling is large facilitating fast radiationless transitions between the surfaces1 c111 c2122 c121 c222HeH11H12H21H22HijiHejH H11 H221 cos21 sin222 sin21 cos22si

4、n2H12H2H122cos2H11 H22H2H122iE1,2H11 H22H2 H1222H11(R)=H22 (R)H12 (R) =0Since two conditions are needed for the existence of a conical intersection the dimensionality is Nint-2, where Nint is the number of internal coordinates For diatomic molecules there is only one internal coordinate and so state

5、s of the same symmetry cannot cross (noncrossing rule). But polyatomic molecules have more internal coordinates and states of the same symmetry can cross.J. von Neumann and E. Wigner, Phys.Z 30,467 (1929)HeH11H12H21H22QyQxQsRrNint-2 coordinates form the seam: points of conicalintersections are conne

6、cted continuouslyhgETwo internal coordinates lift the degeneracy linearly:g-h or branching plane Figure 1b2.933.13.23.33.4r (a.u.)-0.6-0.4-0.200.20.40.6x (a.u.)-3-2-10123E (eV)Figure 4a -0.2-0.100.10.2x (bohr)-0.2-0.100.10.2y (bohr)-0.015-0.01-0.00500.0050.010.015energy (a.u.)H(R) H(R0) H(R0)RH(R) 0

7、 H(R0)RH12(R) 0 H12(R0)RH(R0)R 0H12(R0)R 0g Hh H12He (sxx syy)IgxhyhygxE1,2 sxx syy (gx)2 (hy)2Conical intersections are described in terms of the characteristic parameters g,h,sasymmetrytiltE E0sxx syyg2x2h2y21 cos21 sin222 sin21 cos221( 2) 1()2( 2) 2()TeiA(R)(R;r)(R)gIJ(R)= gI(R) - gJ(R)hIJ(R) cI(

8、Rx)H(R)RcJ(Rx)gI(R) cI(Rx)H(R)RcI(Rx)IecmImm1NCSFHe(R) EI(R)cI(R) 0Locate conical intersections using lagrange multipliers:Eij gjiR 0hjiR 0Additional geometrical constrains, Ki, , can be imposed. These conditions can be imposed by finding an extremum of the Lagrangian. L (R, , )= Ek + 1Eij+ 2Hij + i

9、Ki-2-10123456-4-3-2-101234Y(a0)X(a0)-2-10123456-4-3-2-101234Y(a0)X(a0)hgEFigure 4a -0.2-0.100.10.2x (bohr)-0.2-0.100.10.2y (bohr)-0.015-0.01-0.00500.0050.010.015energy (a.u.)Reaction to H2O+OQuenching to OH(X)+OH(X)OH(A)+OH(X)Three-state conical intersectionsH11(R)=H22 (R)= H33H12 (R) = H13 (R) = H23 (R) =0 Nint-5, where Nint is the number of internal coordinates J. von Neumann and E. Wigner, Phys.Z 30,467 (1929)H H11H12H13H12H22H23H13H23H33H11H120H1T2H12*H22H1T20H11H12*H12H22C.A.Mead J.Chem.Phys., 70, 2276, (1979)1 2 T1 T2 kqEEX)(12111XXkT #p cas(6,6,slaterdet)/6-31G* opt=conical opt