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Quantum Chemistry

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Particle in a box

V=cte



ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑥2 + 𝑉ቇ𝜓ሺ𝑥ሻ= 𝐸𝜓(𝑥)

ቆℎ28𝜋2𝑚 𝛿2𝛿𝑥2 ቇ𝜓ሺ𝑥ሻ= − ሺ𝐸− 𝑉ሻ 𝜓(𝑥)

Ψ(x) = A exp(ikx) + B exp(-ikx)

Ψ(0) = 0 → A = - B

Ψ(L) = 0 → kL = n π (n=1,2,…)



Ψ(x) = A sin(n π x/L)

𝜓ሺ𝑥ሻ= ඨ2𝐿𝑠𝑖𝑛(𝑛𝜋𝑥𝐿 )

𝐸= ℎ28𝑚𝐿2 𝑛2 + 𝑉



Harmonic oscillator

V=½kx2

ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑥2 + 12𝑚𝜔2𝑥2ቇ𝜓ሺ𝑥ሻ = 𝐸 𝜓(𝑥)

k=mω2

E=(n + ½)hν



𝜓𝑛ሺ𝑥ሻ= ට 12𝑛𝑛!ቀ2𝑚𝜔ℎ ቁ

14 𝑒−𝜋𝑚𝜔 𝑥2ℎ 𝐻𝑛(ට2𝜋𝑚𝜔ℎ 𝑥))

𝐻𝑛ሺ𝑥ሻ= (−1)𝑛𝑒𝑥2 𝑑𝑛𝑑𝑥𝑛 𝑒−𝑥2)



Hydrogen atom

V=cte/r



∇2= 𝜕2𝜕𝑟2 + 1𝑟2 𝜕2𝜕𝜃2 + 1𝑟2𝑠𝑖𝑛2𝜃 𝜕2𝜕𝜙2 + 2𝑟 𝜕𝜕𝑟 + cos𝜃𝑟2 sin𝜃 𝜕𝜕𝜃

ቆ− ℎ28𝜋2𝜇∇2 − 𝑒24𝜋𝜀01𝑟ቇ𝜓(𝑟,𝜃,𝜙) = 𝐸𝜓(𝑟,𝜃,𝜙)

𝜓ሺ𝑟,𝜃,𝜙ሻ= 𝑅ሺ𝑟ሻ.𝛩ሺ𝜃ሻ.Φ(𝜙)

ቈ− ℎ28𝜋2𝜇ቆ𝑟2𝑠𝑖𝑛2𝜃 𝜕2𝜕𝑟2 + 𝑠𝑖𝑛2𝜃 𝜕2𝜕𝜃2 + 𝜕2𝜕𝜙2+ 2𝑟𝑠𝑖𝑛2𝜃 𝜕𝜕𝑟 + cos𝜃sin𝜃 𝜕𝜕𝜃ቇ− 𝑒2𝑟𝑠𝑖𝑛2𝜃4𝜋𝜀0 𝜓(𝑟,𝜃,𝜙) = 𝑟2𝑠𝑖𝑛2𝜃𝐸𝜓(𝑟,𝜃,𝜙)



− 𝜕2𝜕𝜙2 𝛷(𝜙) = 𝑚2𝛷(𝜙)

m=…,-2,-1,0,1,2,…𝛷ሺ𝜙ሻ= 1ξ2𝜋𝑒−𝑖𝑚𝜙

𝐿𝑧 = −𝑖 ℎ2𝜋 𝜕𝜕𝜙

𝐿𝑧2𝛷ሺ𝜙ሻ= − ℎ24𝜋2 𝜕2𝜕𝜙2 𝛷ሺ𝜙ሻ= −ℎ2𝑚24𝜋2 𝛷(𝜙)



𝛩ሺ𝜃ሻ= ඨሺ2𝑙 + 1ሻ2 ሺ𝑙 − ȁ4𝑚ȁ4ሻ!

ሺ𝑙 + ȁ4𝑚ȁ4ሻ!𝑃𝑙,𝑚(cos𝜃)

𝑃𝑙,|𝑚|ሺ𝑥ሻ= (1− 𝑥2)|𝑚|/2 𝑑|𝑚|𝑃|𝑚|(𝑥)𝑑𝑥|𝑚|

𝐿2𝛩ሺ𝜃ሻ𝛷ሺ𝜙ሻ= −ℎ2𝑙(𝑙 + 1)4𝜋2 𝛩(𝜃)𝛷(𝜙)



𝑅ሺ𝑟ሻ= ඨ൬2𝑍𝑛𝑎0൰

3 ሺ𝑛− 𝑙 − 1ሻ!2𝑛ሾሺ𝑛+ 𝑙ሻ!ሿ3 𝑒− 𝑟𝑍𝑛𝑎0 ൬2𝑟𝑍𝑛𝑎0൰𝑙 𝐿𝑛−𝑙−12𝑙+1 ൬

2𝑟𝑍𝑛𝑎0൰

𝐿𝑛𝛼ሺ𝑥ሻ= 𝑥− 𝛼𝑒𝑥𝑛! 𝑑𝑛𝑑𝑥𝑛 (𝑒−𝑥𝑥𝑛+𝛼)

𝑎0 = 𝜀0ℎ2𝜋𝑚𝑒2 𝑅𝐻= 𝑚𝑒48ℎ2𝜀02

𝐸= 𝑅𝐻𝑍2𝑛2



n l m Ψnlm=RnlΥlm

1 0 0 1ξ𝜋൬1𝑎0൰3/2 𝑒−𝑟/𝑎0

2 0 0 14ξ2𝜋൬1𝑎0൰3/2

൬2− 𝑟𝑎0൰𝑒−𝑟/𝑎0

2 1 0 14ξ2𝜋൬1𝑎0൰3/2 𝑟𝑎0 𝑒−𝑟/2𝑎0cos (𝜃)

2 1 ±1 14ξ𝜋൬1𝑎0൰3/2 𝑟𝑎0 𝑒−𝑟/2𝑎0sin (𝜃)𝑒±𝑖𝜙

3 0 0 181ξ3𝜋൬1𝑎0൰3/2

ቆ27− 18 𝑟𝑎0 + 2൬𝑟𝑎0൰2ቇ𝑒−𝑟/3𝑎0

3 1 0 181ඨ2𝜋൬1𝑎0൰3/2

൬6− 𝑟𝑎0൰𝑟𝑎0 𝑒−𝑟/3𝑎0cos (𝜃)

3 1 ±1 181ξ𝜋൬1𝑎0൰3/2

൬6− 𝑟𝑎0൰𝑟𝑎0 𝑒−𝑟/3𝑎0sin (𝜃)𝑒±𝑖𝜙

3 2 0 181ξ6𝜋൬1𝑎0൰3/2

൬𝑟𝑎0൰

2 𝑒−𝑟/3𝑎0ሺ3cos2ሺ𝜃ሻ− 1ሻ 3 2 ±1 181ξ𝜋൬1𝑎0൰

3/2൬

𝑟𝑎0൰2 𝑒−𝑟/3𝑎0 sinሺ𝜃ሻcos(𝜃)𝑒±𝑖𝜙

3 2 ±2 1162ξ𝜋൬1𝑎0൰3/2

൬𝑟𝑎0൰

2 𝑒−𝑟/3𝑎0 sin2ሺ𝜃ሻ𝑒±2𝑖𝜙


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