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1 1 Dept of Mechanical Engineering MIT Nanoengineering group
2 » To calculate all the properties of a molecule or crystalline system knowing its atomic information: Atomic species Their coordinates The Symmetry The primitive (unit) cell 2
3 Calculate the ground state many-body wavefunction, and the total energy (r,...,r ),E 1 N Solutions to the MB Schroedinger s equation: H(r,...,r ) (r,...,r ) E (r,...,r ) 1 N 1 N 1 N and eventually the lowest excited states 3
4 Most properties of interest can be expressed as derivatives of the total energy Transport (electrical and thermal conductivity) Magnetic (susceptibility, magnetization) Optical (band gap, dielectric constant, ) Elastic (elastic constants) Thermodynamic (Cv, a, g ) and other RESPONSE FUNCTIONS, which can be expressed as: 2 E/ A B 4
5 » Central quantity defining the ground state» Generalized forces (1 st derivatives) magnetization, charge density: wrt B, V Forces on atoms: wrt atomic displacement Electron affinity and Ionization potential: wrt N» Generalized Susceptibilities (2 nd derivatives) Hardness: wrt Volume Bandgap (chemical hardness): wrt electron number Magnetic susceptibility: wrt B-field 5
6 E(N) EA E(N 1) E(N) IP E(N) E(N 1) E / N (EA IP) / 2 E Gap E / N 2 2 N1 E(N 1) E(N 1) 2E(N) IP EA N -IP -EA N-1 N N+1 Electronegativity (Mulliken) 6
7 » Dynamics: Born-Oppenheimer approximation Due to their fast dynamics compared to ionic motion, electrons stay at all times in their ground state (the BO surface) 7
8 » The many-body wavefunction is a function of 3N coordinates (for an N-electron system) H(r,...,r ) (r,...,r ) E (r,...,r ) 1 N 1 N 1 N» Mean-field idea: one electron is moving in the average field created by N-1 other electrons also called independent electron approximation Reduces the problem to a one-body one with only 3 coordinates in the wavefunction Works at higher dimensions due to reduced quantum fluctuations A A (A A ) f (A) f ( A ) O f "( A ) 2 A 8
9 » No exact solution!» Approximation methods: Variational method An approximation to the ground state can be found by minimizing E var * 0 var guess guess 1 N E E H dr...dr Need to assume a form for the variational wavefunction 9
10 » Reduce the many-body problem to a one-body problem» What is the best mean field? How do we define it?» Guess: Electrostatic field: Hartree Approximation (1927)» Add exchange effects due to fermion statistics (Pauli): Hartree-Fock Approximation (1928)» Density-functional Theory (1964) 10
11 » Hartree: the N-electron WF is a product of oneelectron WFs (r,...,r ) (r )... (r ) 1 N 1 1 N N» Hartree-Fock: the N-electron WF is an antisymmetric product of one-electron WFs 1 (r 1,...,r N ) ( 1) (1) (r 1 )... (N)(r N) N! permuation It can also be written as a Slater determinant, which manifestly satisfies Pauli 11
12 » If the total energy is minimized wrt, with the normality constraint, then the resulting equations satisfied by, are called the Hartree/HF equations. 12
13 * (Evar Constraint) / 0 H (r) E (r) E s are Lagrange multipliers coming from the normalization constraint H 2 p V 2m ion ( r) V Coulomb ( r) V exchange ( r) (r ') (r ') (r ') occupied ' ' V Coulomb (r) dr ' dr ' ' r r ' r r ' (r ') (r ') V (r) (r) dr ' (r) occupied ' exchange ' ' ' r r ' 13
14 Ground state defined by occupied states 14
15 » Underbinds molecules (by typically 10%)» Over-estimates the gap by a factor of 2» Predicts a pseudo-gap for the jellium model Due to a lack of electron screening The exchange potential (or single Slater determinant) does not tell the whole story! We need to add other electronic configurations to the wavefunction to better describe screening 15
16 » Due to Hohenberg and Kohn (1964) and Kohn- Sham (1965) who formulated the variational problem:» If the ground state density is defined as: (r) N (r,r 2,...,r N) dr 2...drN» There is a one to one correspondence between the ground state charge density and the manybody wavefunction. E o is a UNIQUE functional of the ground state density V,E (r,...,r ) (r) ext o o 1 N o 2 16
17 V (r,...,r ) (r) ext o 1 N o» Can two different external potentials produce the same (non-degenerate) ground state density? H T V V Take V1 and V2 leading to H1 and H2 and 1 and 2 H E H E V V but V V (r)[v (r) v (r)]dr o 1 2 likewise E E V V E (r)[v (r) v (r)]dr ee o 2 1 implying E E E E (r,...,r ) V (r) o 1 N ext o ext Eo Min E[ ] E[ o] o [ o] 17
18 » E o is obtained by minimizing this functional E[] wrt with the constraint that (r) dr» Kohn&Sham derived the resulting one-body Schroedinger-like equations bearing their name N 18
19 Solve self consistently occupied H KS[ ] ; (r) (r) 2 2 p H KS[ ] v ext (r) v Coulomb[ (r)] v xc[ (r)] 2m with v [ (r)] E / (r) and xc xc E[ ] T E E E H xc ext 19
20 E[ ] T E E E ext H ext i H xc ext E v (r) (r) dr E 1 (r) (r ') dr dr ' 2 r r ' 2 pi T[ ] and E xc[ ] Vee E H?? 2m 5/3 T TF[ ] C (r) dr E [ ] D (r) dr Jellium 4/3 x Thomas-Fermi-Dirac 20 functional
21 » So in order to get the GROUND STATE energy and electron density, one needs to minimize E[], or in other words, solve the KS equations.» Most physical properties are then obtained from finite differences of this GS energy.» Approximation: the mysterious XC energy functional and the resulting XC potential» Caveat: ONLY o and E o are supposed to be correct! 21
22 » Also proposed by HK (1964) E (r) e (r) dr (r) e [ (r)]dr LDA hom XC XC XC Uses the XC energy density of the Jellium model calculated numerically exactly by QMC: V xc = V xc (» It satisfies the xc-hole sum rule, explaining its success. E (r) e [ (r), (r)]dr with LSDA XC xc LSDA XC XC v (r) E [, ] / (r) 22
23 » Overestimates binding energies ( by 15%)» Underestimates the gap ( by 40%)» Good for solids but bad for molecules Overscreens, since functionals are taken from Jellium model But surprisingly successful in predicting the correct crystal structure and elastic constants (<5% error) for a large variety of systems. 23
24 » GW method: Many-body perturbation theory, take DFT GS as reference unperturbed and the rest of Coulomb interactions as perturbation Solves the screening problem and produces correct band gaps» Quantum Monte Carlo: Very accurate solution to the ground state energy and MB-WF» Configuration-Interaction: Diagonalize the MB Hamiltonian in the basis formed of Slater determinants of occupied and unoccupied states (chemists method of choice, used for molecules) 24
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