Born-Oppenheimer approximation.pptx

Born-Oppenheimer approximation.pptx

• 1.
  Born-Oppenheimer approximation & Slater determinant By -Gokila.N
• 2.
  The Born-Oppenheimer Approximationis the assumption that the electronic motion and the nuclear motion in molecules can be separated . This involves the following assumptions: • The electronic wave function depends upon the nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed.
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  • Electrons aremuch lighter than the nuclei . Hence electrons in a molecule would much more faster than the nuclei • The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the speedy electrons
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  Wave function ψof a molecule is the product of the electronic wave function and the nuclear wave function . ψ = ψe ψN Hamiltonian operator of a molecule Ĥ = TN + Te + VeN + Vee + VNN ------------------ (1) Ĥe = Te + VeN + VNN ------------------ (2) Te + VeN + VNN = Hamiltonian for electron motion VNN = Hamiltonian for nu-nu repulsion ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
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  Ĥ = Ĥe+ VNN ---------------- (3) Consider hydrogen molecule ^
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  Operator for thekinetic energy of the electron T = - ½ ∇1 2 - ½ ∇2 2 Operator for the potential energy due to the electron nuclear attraction VeN = −1 𝑟𝑎1 −1 𝑟b1 −1 𝑟𝑎2 −1 𝑟𝑏2 Operator for the potential energy due to the electron- electron repulsion Vee = 1 𝑟12 ^ ^ ^
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  Operator for thepotential energy due to the nu-nu repulsion VNN = 1 𝑅 Hamiltonian of the molecule Ĥ = −1 𝑟𝑎1 −1 𝑟b1 −1 𝑟𝑎2 −1 𝑟𝑏2 + 1 𝑟12 + 1 𝑅 Ĥ = Ĥe + 1 𝑅 ^
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  Slater determinant In quantummechanics, a Slater determinant is an expression that describes the wave function of a multi- fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons
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  • The Slaterdeterminant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital, where denotes the position and spin of a single electron. • A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero
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  For Helium atom = =1/2! 1𝑠(1)α(1) 1𝑠(2)α(2) 1𝑠(1)β(1) 1𝑠(2)β(2) ψ(1,2) =1/ 2! 1𝑠(1)α(1) 1𝑠(1)β(1) 1𝑠(2)α(2) 1𝑠(2)β(2) -ψ(1,2)
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  For Lithium atom =1/3! 1𝑠(1)α(1) 1𝑠 2 α(2) 1𝑠(3)α(3) 1𝑠(1)β(1) 1𝑠(2)β(2) 1𝑠(3)β(3) 2𝑠(1)α(1) 2𝑠(2)α(2) 2𝑠(3)α(3) ψ(1,2,3)