Computational Physics

38 The variational method for the Schrödinger equation

shall use the following notation:nandmlabel the states inA,αandβlabel the
states inB, andpandqlabel the states in both sets. Furthermore we define

Hpq′ =Hpq( 1 −δpq), (3.37)

that is,H′isHwith the diagonal elements set to 0. Now we can write Eq. (3.11) as

(E−Hpp)Cp=

∑

nA

Hpn′ Cn+

∑

αB

Hp′αCα. (3.38)

If we define
h′pn=Hpn′ /(E−Hpp), (3.39)

and similarly forh′pα, then we can writeEq. (3.38)as

Cp=

∑

nA

h′pnCn+

∑

αB

h′pαCα. (3.40)

Using this expression to rewriteCαin the second term of the right hand side, we
obtain

Cp=

∑

nA

h′pnCn+

∑

αB

h′pα





∑

nA

h′αnCn+

∑

βB

h′αβCβ





=

∑

nA

(

hpn′ +

∑

αB

h′pαh′αn

)

Cn+

∑

αB

∑

βB

h′pαh′αβCβ. (3.41)

After using(3.40)again to re-expressCβand repeating this procedure over and
over, we arrive at

Cp=

∑

nA

(

hpn′ +

∑

αB

h′pαh′αn+

∑

α,βB

h′pαh′αβh′βn+···

)

Cn. (3.42)

We now introduce the following notation:

UpnA =Hpn+

∑

αB

Hp′αHα′n

E−Hαα

+

∑

αβ B

Hp′αHαβ′ Hβ′n

(E−Hαα)(E−Hββ)

+··· (3.43)

Then(3.42)transforms into

Cp=

∑

nA

UpnA −Hpnδpn

E−Hpp

Cn. (3.44)

ChoosingpinA(and calling itm), (3.44) becomes

(E−Hmm)Cm=

∑

nA

UmnACn−HmmCm, (3.45)