Further Improvements Optimization of ls orbitals H πexp(-r) He 2512Texp(-2r) Trial wave function: k32 2 exp(-hr E W,(K,R) at each R choose k so that aW,/ak=0 Results D=2.36eV.R=1.06A Inclusion of other atomic orbitals 2 ResutlsD=2.73eVR=106A
Trial wave function: k 3/2 -1/2 exp(-kr) Eg = W1 (k,R) at each R, choose k so that W1 /k = 0 Results: De = 2.36 eV, Re = 1.06 A Resutls: De = 2.73 eV, Re = 1.06 A 1s 2p Inclusion of other atomic orbitals Further Improvements H -1/2 exp(-r) He+ 2 3/2 -1/2 exp(-2r) Optimization of 1s orbitals
Linear equations 1. two linear equations for two unknown, x, and x aux, aux,=b a21x122202 112212421)1 b,an-b aua 1122a1221)2 x=bau-b, a
a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 (a11a22-a12a21) x1 = b1 a22-b2 a12 (a11a22-a12a21) x2 = b2 a11-b1 a21 Linear Equations 1. two linear equations for two unknown, x1 and x2
Introducing determinant aa 12 =a112 12-21 a 122 a 11a12 12 X a a 21c22 22 1112 a x 21a22
Introducing determinant: a11 a12 = a11a22-a12a21 a21 a22 a11 a12 b1 a12 x1 = a21 a22 b2 a22 a11 a12 a11 b1 x2 = a21 a22 a21 b2
Our case: b,=b,=0, homogeneous 1. trivial solution: x =x2=0 2. nontrivial solution a 012=0 a 2122 n linear equations for n unknown variables ax,t a 1212++a1x a21x1+a2x2++a
Our case: b1 = b2 = 0, homogeneous 1. trivial solution: x1 = x2 = 0 2. nontrivial solution: a11 a12 = 0 a21 a22 n linear equations for n unknown variables a11x1 + a12x2 + ... + a1nxn= b1 a21x1 + a22x2 + ... + a2nxn= b2 ............................................ a x + a x + ... + a x =
aa 1k-101 k+1 n 2k+1 det(a)x n2 nk-12 nk+ nn where a a a In 21a22 det(a)= a. a n nn
a11 a12 ... a1,k-1 b1 a1,k+1 ... a1n a21 a22 ... a2,k-1 b2 a2,k+1 ... a2n det(aij) xk= . . ... . . . ... . an1 an2 ... an,k-1 b2 an,k+1 ... ann where, a11 a12 ... a1n a21 a22 ... a2n det(aij) = . . ... . an1 an2 ... ann