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Cold molecules. Mike Tarbutt. Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules. Lecture 3 – Making cold molecules from cold atoms. Lecture 4 – The Stark shift.
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Cold molecules Mike Tarbutt
Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules. Lecture 3 – Making cold molecules from cold atoms. Lecture 4 – The Stark shift. Lecture 5 – Decelerating, storing and trapping molecules with electric fields. Outline • Review articles: • “Molecule formation in ultracold atomic gases”, J.M. Hutson and P Soldan, International Reviews in Physical Chemistry, 25, 497 (2006) • “Production and application of translationally cold molecules”, H.L. Bethlem and G. Meijer, International Reviews in Physical Chemistry, 22, 73 (2003)
Born-Oppenheimer approximation Nuclear mass ~104 times electronic mass Nuclei move very slowly compared to electrons Solve the electronic Schrodinger equation with “frozen nuclei” Do this for many different values of internuclear separation, R Obtain electronic energies as functions of R – potential energy curves Then solve the Schrodinger equation for the nuclear motion
The Hamiltonian for a diatomic molecule (*non-relativistic) Kinetic energy of nuclei Kinetic energy of electrons Coulomb potential between electrons and nuclei I’ll use subscripts A and B to denote the two nuclei, and the index i to label all the electrons
Let’s immediately simplify the nuclear kinetic energy term using centre-of-mass coordinates and relative coordinates: This transforms the nuclear kinetic energy from to Remove the centre-of-mass motion… We’re interested in the internal energy of the molecule, not the translation of the centre of mass
Need to solve: Electronic Hamiltonian Electronic wave equation and are a set of eigenfunctions and eigenvalues, each corresponding to an electronic state N.B. The eigenfunctions form a complete set at every value of R: Let’s first solve a different problem! Clamp the nuclei in place at a fixed separation R,. The equation to solve is the same, except that the nuclear kinetic energy vanishes. This electronic wave equation can be solved using the same techniques as in the atomic case (e.g. Hartree-Fock)
Nuclear wavefunctions Electronic wavefunctions Substitute this expansion into the full Schrodinger equation, …and use the result Multiply by , integrate over electronic coordinates, and use the orthonormality condition: Let’s now expand the complete wavefunction on the basis of the electronic eigenfunctions: …an infinite set of coupled differential equations which determine the nuclear wavefunctions
Nuclear kinetic energy Effective potential for nuclear motion Adiabatic approximation - the nuclear motion does not mix the electronic states – then the set of equations uncouple: Simplify... Neglect the 2nd and 3rdterms (it turns out they are very small)... Then we obtain a much simpler equation for the nuclear wavefunction:
Separate the nuclear wave equation… Central potential – Separate into radial and angular parts: As always for a central potential, the angular functions are spherical harmonics: They are eigenfunctions of J2 and Jz Then we’re left with a fairly simple looking radial equation…
Solve the radial equation… where Define Ev such that the total energy is Could solve this numerically, or find an approximate solution for R close to R0 • Set R=R0 in the denominator of the second term. 2) Expand En(R) in a Taylor series about R0 We’re left with a harmonic oscillator equation:
Summary The total energy is Solve the electronic wave equation with fixed nuclei Repeat for many different R to obtain potential curves These appear as the potential in the nuclear wave equation Can separate and (approximately) solve this wave equation The total wavefunctions are products of electronic, vibrational and rotational eigenstates
Energy scales Electronic frequencies ~ 1015 Hz Vibrational frequencies ~ 5 1013 Hz Rotational frequencies ~ 1010 - 1012 Hz
Things we’ve left out... • Coupling terms neglected in the Born-Oppenheimer approximation • Centrifugal distortion • Spin-orbit interaction • Spin-rotation interaction • Spin-spin interaction • Lambda doubling • Magnetic hyperfine interactions • Electric quadrupole interactions
Some molecular notation Remember how it goes for an atom: (configuration) 2S+1LJ e.g. Ground state of sodium: 1s22s22p63s 2S1/2 What are the “good quantum numbers” for a diatomic molecule (which operators commute with the Hamiltonian)? The lack of spherical symmetry in a diatomic molecule means that L2 does not commute with H. However, to a good approximation, Lz does commute with H when the z-axis is taken along the internuclear axis. Therefore, the projection of the total orbital angular momentum onto the internuclear axis is (approximately) a good quantum number – labelled by L, which can be S, P, D... Spin multiplicity X, A, B, C... The electronic states of a molecule are labelled: (unique letter) 2S+1LW Projection of orbital angular momentum onto internuclear axis Projection of total angular momentum onto internuclear axis Can also include the vibrational state v, and the total angular momentum J e.g. one of the excited states of CaF is written: A 2P3/2 (v=2, J=7/2)
