The Projector

We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

As has been shown previously [243], both sets can be described by eigenvalue equations, but for the set 2 it is more direct to work with projectors Pr taking the values 1 or 0. Let us consider a class of functions/(x), describing the state of the system or a process, such that (for reasons rooted in physics)/(x) should vanish for X D (i.e., for supp/(x) = D, where D can be an arbifiary domain and x represents a set of variables). If Pro(x) is the projector onto the domain D, which equals 1 for x G D and 0 for x D, then all functions having this state property obey an equation of restriction [244] ... 

For functions of a single variable (e.g., energy, momentum or time) the projector Prz)(x) is simply 0(a ), the Heaviside step function, or a combination thereof. When also replacing x, k by the variables , t, the Fourier transform in Eq. (5) is given by... 

We can use this matrix to project onto the component of a vector in the v(i) direction. For the example we have been considering, if we form the projector onto the v(l) vector, we obtain... 

These equations state that the three Ish orbitals can be combined to give one Ai orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors... 

This shows that Po is indeed the projector onto the non-relativistic eigenstates of energy Eo-In conclusion, ffo = foTFo/oandffi = Po7 iPo + fof i7t o/o + fo7Fof ifo,arezeroinEq.(6). 

Spot size. The size of the LGS is a critical issue, since it dehnes the saturation effects of the laser and the power needed to reach a given system performance, and also the quality of the wavefront sensing. There is an optimum diameter of the projector, because, if the diameter is too small, the beam will be spread out by diffraction and if it is too large it will be distorted due to atmospheric turbulence. The optimum diameter is about 3ro, thus existing systems use projection telescopes with diameters in the range of 30-50 cm. 

Figure 12. Schematic view of the projector used to launch 3 simultaneous Rayleigh beams into the sky. A spinning mirror Ml at an image of the exit pupil directed successive laser pulses into three separate beams (only 2 shown here) without causing beam wander on the primary mirror M2. Electronic synchronization was implemented between the motor and the laser.

A camera to look down through the projector for alignment of L4. 

The projector chosen for Keck is an afocal Gregorian telescope (see Ch. 3) which expands the beam to a full 50-cm round profile. The final lens is placed with the plane surface facing the sky and the aspheric surface ground to give a 1/10 wave (rms) transmitted wave front. A back reflection from the last surface is used to monitor the quality of the wave front as it leaves the dome. 

In the atomic case, the pwc s are defined by the ion level I and the / value of the electron partial wave, i.e. the formal pwc subsets span the tensor product of the ion level states times the one-electron /-wave manifold. In the following, the subspaces will be indexed with greek letters a subspace index a=/ ,/ will designate explicitly an open pwc subspace, while an index 0 an arbitrary subspace. The Ic subspace will be numbered 0 and Qp will denote the projector in the subspace 0. 

Kresse, G. and Joubert, D. (1999) From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B - Condensed Matter, 59, 1758-1775. 

Blochl, P. E., Margl, P, Schwarz, K., 1996, Ab Initio Molecular Dynamics With the Projector Augmented Wave Method in Chemical Applications of Density Functional Theory, Laird, B. B., Ross, R. B., Ziegler, T. (eds.), American Chemical Society, Washington DC. 

We proceed as follows for a complex P, we count the number of real parameters which completely define the entire matrix from this number, we subtract the number of real conditions imposed by TV-representability (i.e. hermiticity, rank N and unit eigenvalues). The remaining number of parameters represents the number of real (experimental) conditions required to complete the definition of the projector considered. Such a number is the solution to the problem posed in this paper. Later on, we shall consider the two other cases previously mentioned, that is, complex independent parameters of a complex , and real independent parameters of a real . 

By simply using arguments analogous to those used in deriving Equation (28), one may find that if the projector is real, then... 

Pecora noticed that the phase information of C is lost in the original constraints [i.e. P2 = P Tr P = N], but found it not at all clear . Here, we showed in which way one might take into account the loss of the phase information in C when calculating the number of conditions to uniquely determine C one has to impose, over and above the constraints arising from fixing the projector, the conditions to determine a particular unitary transformation in the TV-dimensional subspace, apart from the phases of the basis functions which are physically meaningless in the context of Quantum Mechanics. 

Hence, for finite times t any initial vector from 77 remains in this subspace up to an error of the order h°°. Moreover, for semiclassically large times, t 0, the error is still approaching zero in the semiclassical limit. Since the construction of the projectors fl was based on the classical projection matrices (6) associated with positive and negative... 

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