inhomogeneous case: b,t 0 for at least one k a 1,k-11 1.k+1 n a a21b2 e. k+ 2n an1an…ankb2ank1…an nn det( a
a11 a12 ... a1,k-1 b1 a1,k+1 ... a1n a21 a22 ... a2,k-1 b2 a2,k+1 ... a2n . . ... . . . ... . an1 an2 ... an,k-1 b2 an,k+1 ... ann xk = det(aij) inhomogeneous case: bk = 0 for at least one k
homogeneous case: bk =0,k=1,2,.,n (a) traval case: X=0,k=1,2.,n (b)nontravial case: det(a; )=0 For a n-th order determinant det(ai )=2ak cik where, Clk is called cofactor
(a) travial case: xk = 0, k = 1, 2, ... , n (b) nontravial case: det(aij) = 0 homogeneous case: bk = 0, k = 1, 2, ... , n For a n-th order determinant, n det(aij) = alk Clk l=1 where, Clk is called cofactor
Trial wave function o is a variation function Which is a combination of n linear independent functions{f1,2,…f n)5 φ=c1+c2+…+crn →2[(Hk-S1W)ck]=01=1,2…,n H=∫dfHf W=∫dbH/drp
Trial wave function f is a variation function which is a combination of n linear independent functions { f 1 , f 2 , ... f n }, f = c1 f 1 + c2 f 2 + ... + cn f n n [( Hik - SikW ) ck ] = 0 i=1,2,...,n k=1 Sik dt f i f k Hik dt f i H fk W dt f H f / dt f f
Linear variational theorem (1)W1≤W2s….≤ wn are n roots of Eq(1)2 (1)E1≤E2≤….≤En≤En+1s…. are energies of eigenstates then,W1≥E1,W2≥E2,…,Wn≥En
(i) W1 W2 ... Wn are n roots of Eq.(1), (ii) E1 E2 ... En En+1 ... are energies of eigenstates; then, W1 E1 , W2 E2 , ..., Wn En Linear variational theorem
Molecular Orbital(MO): P=C,V1+ C,y (H1-W)c1+(H12-SW)c2=0 S1=1 (H21-SW)c1+(H2-W)c2=0 Generally: iv i a set of atomic orbitals, basis set LCAO-MOφ=c11+C2V2+ ●@●●●● +c nrn linear combination of atomic orbitals ∑(H1-SW)ck=01=1,2,……n hi dty Hv Sk=dtV W Sk-1
Molecular Orbital (MO): f = c1y1 + c2y2 ( H11 - W ) c1 + ( H12 - SW ) c2 = 0 S11=1 ( H21 - SW ) c1 + ( H22 - W ) c2 = 0 S22=1 Generally : yi a set of atomic orbitals, basis set LCAO-MO f = c1y1 + c2y2 + ...... + cn fn linear combination of atomic orbitals n ( Hik - SikW ) ck = 0 i = 1, 2, ......, n k=1 Hik dt yi * H yk Sik dt yi * yk Skk = 1