{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

1PcET8-lecture3

1PcET8-lecture3 - Quantum Chemistry Theory Computational...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Quantum Chemistry Theory Computational Chemistry Ab initio methods seek to solve the Schr¨ odinger equation. Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for the electron density function. Semi-empirical methods use approximate Hamiltonians that are partly parameterized using experimental data. Molecular mechanics uses Newtonian mechanics and empirically parameterized force fields. Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 1 / 56
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Quantum Chemistry Theory The Time-Dependent Schr¨ odinger Equation The time-dependent Schr¨ odinger equation can be written as i Ψ ( x , t ) t = H Ψ ( x , t ) where = 1 . 055 × 10 34 J s is the reduced Planck’s constant; the operator H is the system Hamiltonian; Ψ ( · , t ) is the wavefunction of the system at time t . The modulus of the wavefunction, | Ψ ( · , t ) | 2 , is often interpreted as the probability density of the system at time t . Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 2 / 56
Image of page 2
Quantum Chemistry Theory The Time-Independent Schr¨ odinger Equation If H is time-indepedent, then we can seek an eigenfunction expansion Ψ ( x , t ) = n =1 c n Ψ n ( x ) e i En t where Ψ n satisfies the eigenvalue problem H Ψ n = E n Ψ n and E n is interpreted as the energy of the stationary configuration Ψ n . The eigenfunction Ψ 0 corresponding to the smallest eigenvalue E 0 is called the ground state of the system. Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 3 / 56
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Quantum Chemistry Theory The Hamiltonian of a Molecule The Hamiltonian H can be written as the sum of the kinetic and potential energy operators H = T + V , where for a (non-relativistic) molecule T = 2 2 i 1 m i 2 x 2 i + 2 y 2 i + 2 z 2 i V = 1 4 π 0 i < j e i e j r ij (electrostatic potential) . Here i , j range over nuclei and electrons, m i and e i are the mass and charge of the i ’th particle, and 0 is the vacuum permittivity. Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 4 / 56
Image of page 4
Quantum Chemistry Hydrogen-like atom Hydrogen-like Atom In a few cases, Schr¨ odinger’s equation can be solved analytically. For a single nucleus with mass M and charge + Ze with a single electron (of mass m ), we have: 2 2( M + m ) 2 CM 2 2 μ 2 Ze 2 4 π 0 r Ψ = E Ψ , where 2 CM is the Laplacian for the center-of-mass coordinates R CM = M r a + m r e M + m 2 is the Laplacian for the coordinates of the electron relative to the nucleus: r = r e r a μ is the reduced mass μ = ( M 1 + m 1 ) 1 m . Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 5 / 56
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Quantum Chemistry Hydrogen-like atom Reduction of Dimension Because the potential energy depends only on the distance of the electron to the nucleus, the center-of-mass can be separated out of the previous equation, leaving 2 2 μ 2 Ψ Ze 2 4 π 0 r Ψ = E Ψ where r = | r | . Transforming to polar coordinates ( r , θ , φ ) gives 2 2 m 1 r 2 sin( θ ) sin( θ ) r r 2 r + ∂θ sin( θ ) ∂θ + 1 sin( θ ) 2 ∂φ 2 Ψ Ze 2 4 π 0 r Ψ = E Ψ .
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern