Physics “Advanced Electronic Structure” LMTO family: ASA, Tight-Binding and Full Potential Methods Contents: 1. ASA-LMTO equations 2. Tight-Binding LMTO Method 3. Full Potential LMTO Method 4. Advanced Topics: exact LMTOs, NMTOs.

Linear Muffin-Tin Orbitals Consider envelope function as Inside every sphere perform smooth augmentation which gives LMTO construction.

LMTO definition (  dependence is highlighted): Variational Equations which should be used as a basis in expanding Variational principle gives us matrix eigenvalue problem.

Integrals between radial wave functions: LMTO Hamiltonian one-center integrals two-center integrals three center integrals

How linearization works: KKR-ASA equations produce energy dependent orbitals Energy dependence can be factorized to linear order As a result, a prefactor  (D) can be combined with the variational coefficients A! Therefore, energy dependence of the orbitals cancels out with the normalization.

Energy dependent MTO in KKR-ASA method Keeping in mind the use of variational principle (not tail cancellation condition) let us augment the tails

Energy dependent MTO in KKR-ASA method becomes Perform linearization: is the function having given logarithmic derivative D: We obtain

Then one can show that the energy dependence hidden in D(E) factorizes where LMTOs are given by (kappa=0 approximation) Or for any fixed kappa:

Accuracy of LMTOs: LMTO is accurate to first order with respect to (E-E ) within MT spheres. LMTO is accurate to zero order (  2 is fixed) in the interstitials. Atomic sphere approximation: Blow up MT-spheres until total volume occupied by spheres is equal to cell volume. Take matrix elements only over the spheres. Multiple-kappa LMTOs: Choose several Hankel tails, typically decaying as 0,-1,-2.5 Ry. Frequently, double kappa (tripple kappa) basis is sufficient.

Tight-Binding LMTO Tight-Binding LMTO representation (Andersen, Jepsen 1984) LMTO decays in real space as Hankel function which depends on  2 =E-V 0 and can be slow. Can we construct a faster decaying envelope? Advantage would be an access to the real space hoppings, perform calculations with disorder, etc:

Tight-Binding LMTO Any linear combination of Hankel functions can be the envelope which is accurate for MT-potential where A matrix is completely arbitrary. Can we choose A-matrix so that screened Hankel function is localized? Electrostatic analogy in case  2 =0 Outside the cluster, the potential may indeed be screened out. The trick is to find appropriate screening charges (multipoles)

Screening LMTO orbitals: Unscreened (bare) envelopes (Hankel functions) Screening is introduced by matrix A Consider it in the form where alpha and S_alpha coefficents are to be determined.

We obtain when r is within a sphere centered at R’’ R R’ R’’ r r-R r-R’ r-R’’ =

Demand now that we obtain one-center like expansion for screened Hankel functions where S_alpha plays a role of (screened) structure constants and we introduced screened Bessel functions

Screened structure constants are short ranged: For s-electrons, transofrming to the k-space Choosing alpha to be negative constant, we see that it plays the role of Debye screening radius. Therefore in the real space screened structure constants decay exponentially while bare structure constants decay as

Screening parameters alpha have to be chosen from the condition of maximum localization of the structure constants in the real space. They are in principle unique for any given structure. However, it has been found that in many cases there exist canonical screened constants alpha (details can be found in the literature). Since in principle the condition to choose alpha is arbitrary we can also try to choose such alpha’s so that the resulting LMTO becomes (almost) orthogonal! This would lead to first principle local-orbital orthogonal basis. In the literature, the screened mostly localized representation is known as alpha-representation of TB-LMTOs. The Representaiton leading to almost orthogonal LMTOs is known as gamma-representation of TB-LMTOs. If screening constants =0, we return back to original (bare/unscreened) LMTOs

Tight-Binding LMTO Since mathematically it is just a transformation of the basis set, the obtained one-electron spectra in all representations (alpha, gamma) are identical with original (long-range) LMTO representation. However we gain access to short-range representation and access to hopping integrals, and building low-energy tight-binding models because the Hamiltonian becomes short-ranged:

Advanced Topics: FP-LMTO Method Problem: Representation of density, potential, solution of Poisson equation, and accurate determination of matrix elements with LMTOs defined in whole space as follows

FP-LMTO Method Ideas: Use of plane wave Fourier transforms Weirich, 1984, Wills, 1987, Bloechl, 1986, Savrasov, 1996 Use of atomic cells and once-center spherical harmonics expansions Savrasov & Savrasov, 1992 Use of interpolation in interstitial region by Hankel functions Methfessel, 1987

At present, use of plane wave expansions is most accurate To design this method we need representation for LMTOs

Problem: Fourier transform of LMTOs is not easy since Solution: Construct psuedoLMTO which is regualt everywhere and then perform Fourier transformation

The idea is simple – replace the divergent part inside the spheres by some regular function which matches continuously and differentiably. What is the best choice of these regular functions? The best choice would be the one when the Fourier transform is fastly convergent. The smoother the function the faster Fourier transform.

Weirich proposed to use linear combinations This gives Wills proposed to match up to n th order This gives with optimum n found near 10 to 12

Another idea ( Savrasov 1996, Methfessel 1996 ) Smooth Hankel functions Parameter  is chosen so that the right-hand side is nearly zero when r is outside the sphere. Solution of the equation is a generalized error-like function which can be found by some recurrent relationships. It is smooth in all orders and gives Fourier transform decaying exponentially

Finally, we developed all necessary techniques to evaluate matrix elements where we have also introduced pseudopotential

Exact LMTOs LMTOs are linear combinations of phi’s and phi-dot’s inside the spheres, but only phi’s (Hankel functions at fixed  ) in the interstitials. Can we construct the LMTOs so that they will be linear in energy both inside the spheres and inside the interstitials (Hankels and Hankel-dots)? Yes, Exact LMTOs are these functions! Advanced Topics

Let us revise the procedure of designing LMTO: Step 1. Take Hankel function (possibly screened) as an envelope. Step 2. Replace inside all spheres, the Hankel function by linear combinations of phi’s and phi-dot’s with the condition of smooth matching at the sphere boundaries. Step 3. Perform Bloch summation.

Design of exact LMTO (EMTO): Step 1. Take Hankel function (possibly screened) as an envelope. Step 2. Replace inside all spheres, the Hankel function by only phi’s with the continuity condition at the sphere boundaries. The resulting function is no longer smooth!

Step 3. Take energy-derivative of the partial wave So that it involves phi-dot’s inside the spheres and Hankel-dot’s in the interstitials. Step 4. Consider a linear combination where matrix M is chosen so that the whole construction becomes smooth in all space (kink-cancellation condition) This results in designing Exact Linear Muffin-Tin Orbital.

Non-linear MTOs (NMTOs) Do not restrict ourselves by phi’s and phi-dot’s, continue Tailor expansion to phi-double-dot’s, etc. In fact, more useful to consider just phi’s at a set of additional energies, instead of dealing with energy derivatives. This results in designing NMTOs which solve Schroedinger’s equation in a given energy window even more accurately. Advanced Topics

MINDLab Software

Understanding s-electron band structure. Tight-binding parameterization for Na. s level position and hopping rate for s electrons.

Understanding s-d electron band structure. Tight-binding parameterization for Cu. s level position and hopping rate for s electrons. d level position and narrow d-bands. s-d hybridization