The value of S12 is calculated as follows:
S12=e−R/a0(1+Ra0+R23a02) …(1)
Where,
• a0 is a Bohr’s radius 0.529A∘.
• R is the distance between two nuclei.
The value of H12 is calculated as follows:
H12=K[a0S12R−e−R/a0(1+Ra0−S122)] …(2)
Where,
• K is the conversion factor from atomic to SI units.
The value of E1 and E2 are given by the formula,
E1=H11+H121+S12E2=H22−H121−S12 …(3)
The wavefunction ϕH2+,1 and ϕH2+,2 is given as follows:
ϕH2+,1=12+2S12(ΨH(1)+ΨH(2))ϕH2+,2=12−2S12(ΨH(1)−ΨH(2)) …(4)
The value of R=1.00 A∘, K=27.09 eV and H11=−0.258 eV is given.
Substitute the value of R and K in the equation (1).
S12=e−1.00 A∘/0.529A∘(1+1.00 A∘0.529A∘ +(1.00 A∘)23(0.529A∘)2)=0.151(1+1.89+1.194)=0.151×4.084=0.616
Thus, the value of S12 is 0.616.
Substitute the value of R, S12 and K in the equation (2).
H12=27.09 eV[0.529A∘×0.6161.00 A∘−e−1.00 A∘/0.529A∘(1+1.00 A∘0.529A∘−0.6162)]=27.09eV[0.325−0.151(1+1.89−0.308)]=27.09eV[0.325−0.151×2.582]=−1.73 eV
Thus, the value of H12 is −1.73 eV.
Substitute the value of H12, S12 and H11 in the equation (3).
E1=−0.258+(−1.73)1+0.616=−1.230 eVE2=−0.258−(−1.73)1−0.616=3.833 eV
Thus, the value of E1 and E2 are −1.230 eV and 3.833 eV.
Substitute the value of S12 in the equation (4).
ϕH2+,1=12+2×0.616(ΨH(1)+ΨH(2))=0.558(ΨH(1)+ΨH(2))ϕH2+,2=12−2×0.616(ΨH(1)−ΨH(2))=1.141(ΨH(1)−ΨH(2))
Thus, the wavefunction ϕH2+,1 and ϕH2+,2 is 0.558(ΨH(1)+ΨH(2)) and 1.141(ΨH(1)−ΨH(2)).
The value of R=1.15 A∘, K=27.09 eV and H11=−0.258 eV is given.
Substitute the value of R and K in the equation (1).
S12=e−1.15 A∘/0.529A∘(1+1.15 A∘0.529A∘ +(1.15 A∘)23(0.529A∘)2)=0.113(1+2.17+1.57)=0.535
Thus, the value of S12 is 0.535.
Substitute the value of R, S12 and K in the equation (2).
H12=27.09 eV[0.529A∘×0.5351.15 A∘−e−1.15 A∘/0.529A∘(1+1.15 A∘0.529A∘−0.5352)]=27.09eV[0.246−0.113(1+2.173−0.267)]=−2.22 eV
Thus, the value of H12 is −2.22 eV.
Substitute the value of H12, S12 and H11 in the equation (3).
E1=−0.258+(−2.22 eV)1+0.535=−1.61 eVE2=−0.258−(−2.22 eV)1−0.535=4.21 eV
Thus, the value of E1 and E2 are −1.61 eV and 4.21 eV.
Substitute the value of S12 in the equation (4).
ϕH2+,1=12+2×0.535(ΨH(1)+ΨH(2))=0.57(ΨH(1)+ΨH(2))ϕH2+,2=12−2×0.535(ΨH(1)−ΨH(2))=1.03(ΨH(1)−ΨH(2))
Thus, the wavefunction ϕH2+,1 and ϕH2+,2 is 0.57(ΨH(1)+ΨH(2)) and 1.03(ΨH(1)−ΨH(2)).
The value of R=1.45 A∘, K=27.09 eV and H11=−0.258 eV is given.
Substitute the value of R and K in the equation (1).
S12=e−1.45 A∘/0.529A∘(1+1.45 A∘0.529A∘ +(1.45 A∘)23(0.529A∘)2)=0.064(1+2.74+2.50)=0.399
Thus, the value of S12 is 0.399.
Substitute the value of R, S12 and K in the equation (2).
H12=27.09 eV[0.529A∘×0.3991.45 A∘−e−1.45 A∘/0.529A∘(1+1.45 A∘0.529A∘−0.3992)]=27.09eV[0.145−0.064(1+2.741−0.1995)]=−2.19 eV
Thus, the value of H12 is −2.19 eV.
Substitute the value of H12, S12 and H11 in the equation (3).
E1=−0.258+(−2.19)1+0.399=−1.75 eVE2=−0.258−(−2.19)1−0.399=3.21 eV
Thus, the value of E1 and E2 are −1.75 eV and 3.21 eV.
Substitute the value of S12 in the equation (4).
ϕH2+,1=12+2×0.399(ΨH(1)+ΨH(2))=0.59(ΨH(1)+ΨH(2))ϕH2+,2=12−2×0.399(ΨH(1)−ΨH(2))=0.91(ΨH(1)−ΨH(2))
Thus, the wavefunction ϕH2+,1 and ϕH2+,2 is 0.59(ΨH(1)+ΨH(2)) and 0.91(ΨH(1)−ΨH(2)).
The value of R=1.60 A∘, K=27.09 eV and H11=−0.258 eV is given.
Substitute the value of R and K in the equation (1).
S12=e−1.60 A∘/0.529A∘(1+1.60 A∘0.529A∘ +(1.60 A∘)23(0.529A∘)2)=0.048(1+3.02+3.05)=0.339
Thus, the value of S12 is 0.339.
Substitute the value of R, S12 and K in the equation (2).
H12=27.09 eV[0.529A∘×0.3391.60 A∘−e−1.60 A∘/0.529A∘(1+1.60 A∘0.529A∘−0.3392)]=27.09eV[0.112−0.048(1+3.02−0.16)]=−1.977 eV
Thus, the value of H12 is −1.977 eV.
Substitute the value of H12, S12 and H11 in the equation (3).
E1=−0.258+(−1.977 eV)1+0.339=−1.66 eVE2=−0.258−(−1.977 eV)1−0.339=2.60 eV
Thus, the value of E1 and E2 are −1.66 eV and 2.60 eV.
Substitute the value of S12 in the equation (4).
ϕH2+,1=12+2×0.339(ΨH(1)+ΨH(2))=0.61(ΨH(1)+ΨH(2))ϕH2+,2=12−2×0.339(ΨH(1)−ΨH(2))=0.87(ΨH(1)−ΨH(2))
Thus, the wavefunction ϕH2+,1 and ϕH2+,2 is 0.61(ΨH(1)+ΨH(2)) and 0.87(ΨH(1)−ΨH(2)).
The values of potential energy and R is given below.
R(Ao) |
Energy (eV) |
1.00 |
3.83 |
1.15 |
4.21 |
1.32 |
16.23 |
1.45 |
3.21 |
1.60 |
2.60 |
The plot between the potential energy with R is shown below.
Figure 1
The values of potential energy and R is given below.
R(Ao) |
Energy (eV) |
1.00 |
−1.23 |
1.15 |
−1.61 |
1.32 |
−6.37 |
1.45 |
−1.75 |
1.60 |
−1.66 |
The plot between the potential energy with R is shown below.
Figure 2