HYDROGENMOLECULE ION

-e

+e

+e

r1

r2

R

HYDROGENMOLECULE ION

𝐻 = −ℏ2

2𝑚𝛻2 −

𝑒2

4𝜋𝜖0(1

𝑟1+

1

𝑟2)

𝜓 = 𝐴 𝜓0 𝑟1 − 𝜓0 𝑟2

𝜓0 𝑟 =1

𝜋𝑎3𝑒− 𝑟 𝑎

1 = 𝜓−2𝑑3𝑟

1 = 𝐴 2 𝜓0(𝑟12𝑑3𝑟 + 𝜓0(𝑟2

2𝑑3𝑟 − 2 𝜓0(𝑟1 𝜓0(𝑟2 𝑑3𝑟

1 1𝜓0

I

𝐴 2 =1

2(1 − 𝐼

𝐼 = 𝜓0(𝑟1 𝜓0(𝑟2 𝑑3𝑟

=1

𝜋𝑎3 𝑒−(𝑟1+𝑟2 𝑎 𝑑3𝑟

𝒓𝟐 = 𝒓𝟐 + 𝑹𝟐 − 𝟐𝒓𝑹𝒄𝒐𝒔𝜽

𝒓𝟏 = 𝒓

𝑹

𝐼 =1

𝜋𝑎3 𝑒− 𝑟 𝑎𝑒− 𝑟2+𝑅2−2𝑟𝑅𝑐𝑜𝑠𝜃 𝑎 𝑟2𝑠𝑖𝑛𝜃𝑑𝑟𝑑𝜃𝑑𝜙

𝜙 2𝜋

𝜃 𝑦 = 𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠𝜃. 𝑑 𝑦2 = 2𝑦𝑑𝑦 = 2𝑟𝑅𝑠𝑖𝑛𝜃𝑑𝜃

0

𝜋

𝑒− 𝑟2+𝑅2−2𝑟𝑅𝑐𝑜𝑠𝜃 𝑎 𝑠𝑖𝑛𝜃𝑑𝜃

𝜃 = 𝜋

𝜃 = 0 𝑦 = 𝑟 − 𝑅

𝑦 = 𝑟 + 𝑅=

1

𝑟𝑅 𝑟−𝑅

𝑟+𝑅

𝑒− 𝑦 𝑎 𝑦𝑑𝑦

= −𝑎

𝑟𝑅𝑒−

𝑟−𝑅𝑎 𝑟 + 𝑅 + 𝑎 − 𝑒−

𝑟−𝑅𝑎 ( 𝑟 − 𝑅 + 𝑎

0

∞

𝑒−𝑘𝑥𝑥𝑛 𝑑𝑥 =𝑛!

𝑘𝑛

𝐼 =2

𝑎2𝑅−𝑒− 𝑅 𝑎

0

∞

𝑟 + 𝑅 + 𝑎 𝑒−2𝑟𝑎 𝑟𝑑𝑟 + 𝑒−

𝑅𝑎

0

𝑅

𝑅 − 𝑟 + 𝑎 𝑟𝑑𝑟 + 𝑒 𝑅 𝑎 𝑅

∞

𝑟 − 𝑅 + 𝑎 𝑒−2 𝑟 𝑎𝑟𝑑𝑟

𝐼 = 𝑒− 𝑅 𝑎 1 +𝑅

𝑎+1

3

𝑅

𝑎

2

𝐻 = −ℏ2

2𝑚𝛻2 −

𝑒2

4𝜋𝜖0(1

𝑟1+

1

𝑟2)

ö 𝐻𝜓 = 𝐸𝜓

−ℏ2

2𝑚𝛻2 −

𝑒2

4𝜋𝜖0𝑟1,2𝜓0 𝑟1,2 = 𝐸1𝜓0(𝑟1,2)

𝐻𝜓− = 𝐴 −ℏ2

2𝑚𝛻2 −

𝑒2

4𝜋𝜖0(1

𝑟1+1

𝑟2𝜓0 𝑟1 − 𝜓0 𝑟2 = 𝐸1𝜓 − 𝐴(

𝑒2

4𝜋𝜖0 𝜓0 𝑟1𝑟1

−𝜓0 𝑟2𝑟2

𝜓0(𝑟1,2

𝐻 = 𝐸1 − 𝑎 𝐴 2(𝑒2

4𝜋𝜖0 𝜓0 𝑟1

1

𝑟2𝜓0 𝑟1 − 𝜓0 𝑟1

1

𝑟1𝜓0 𝑟2

𝐷 = 𝜓0 𝑟1𝑎

𝑟2𝜓0 𝑟1

𝑋 = 𝜓0 𝑟1𝑎

𝑟1𝜓0 𝑟2

𝐷 =𝑎

𝑅− (1 +

𝑎

𝑅 𝑒−2 𝑅 𝑎

𝑋 = (1 +𝑅

𝑎 𝑒− 𝑅 𝑎

𝐻 = 1 + 2𝐷 − 𝑋

1 − 𝐼𝐸1

𝐼 = 𝑒− 𝑅 𝑎 1 +𝑅

𝑎+1

3

𝑅

𝑎

2

𝑉𝑝𝑝 =𝑒2

4𝜋𝜖0

1

𝑅= −

2𝑎

𝑅𝐸1

𝐸𝑡𝑜𝑡 = 𝐻 + 𝑉𝑝𝑝

𝐸𝑡𝑜𝑡 = 1 + 2𝐷 − 𝑋

1 − 𝐼−2𝑎

𝑅𝐸1

𝐹 𝑥 =𝐸𝑡𝑜𝑡−𝐸1

𝐹 𝑥 = −1

𝐸11 + 2

𝐷 − 𝑋

1 − 𝐼−2𝑎

𝑅𝐸1 =

2𝑎

𝑅− 1 − 2

𝐷 − 𝑋

1 − 𝐼

𝐹 𝑥 = −1 +2

𝑥− 2

1𝑥 − 1 +

1𝑥 𝑒−2𝑥 − (1 + 𝑥 𝑒−𝑥

1 − 𝑒−𝑥(1 + 𝑥 +𝑥2

3

= −1 +2

𝑥

1𝑥 − 1 + 𝑥 +

𝑥2

3 𝑒−𝑥 − 1 + 𝑥 + 1 𝑒−2𝑥 + (𝑥 + 𝑥2 𝑒−𝑥

1 − 𝑒−𝑥(1 + 𝑥 +𝑥2

3

𝐹 𝑥 = −1 +2

𝑥

2𝑥2

3− 1 𝑒−𝑥 + (1 + 𝑥 𝑒−2𝑥

1 − 𝑒−𝑥(1 + 𝑥 +𝑥2

3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 1 2 3 4 5 6 7 8 9

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