Haut PDF Coordinates, modes and maps for the density functional

Coordinates, modes and maps for the density functional

Coordinates, modes and maps for the density functional

and associated modes w α which design convenient sets of coordinates and convenient parametrizations of F, δF/δρ, δ 2 F/(δρ δρ ′ ), ... etc., ii) the explicit relation we showed between these modes and the traditional perturbation theories used in the many-body problem and iii) the likely possibility of truncated descriptions and accurate parametrizations. Actually, in the context of extended systems, the idea of density waves as important modes of the system has always been present. There remains to be seen, obviously, if our modes can be generalized to infinite systems. It also remains to be seen whether, for finite or infinite systems, our truncations are always justified, whether infinite resummations are possible, whether collective degrees of freedom are present, or absent, because of our new representation. Also, because of our choice of a Gaussian weight for the new family {Γ}, the present results are better meaningful if restricted to nuclei. A generalization to atoms and molecules obviously demands other weights for the constrained polynomials. The usual perturbation theories have their hierarchy of modes in many-body space, most often a hierarchy of particle- hole components. Our approach, tuned to the one-body nature of the density functional, replaces the particle-holes by other modes, in a transparent way.
En savoir plus

16 En savoir plus

Constrained Density Functional Theory

Constrained Density Functional Theory

function of nuclear position and as such are at the core of many qualitative pictures of molec- ular electronic structure, including the charge-transfer states of section 3 and the uncoupled spin states of section 4. But to fill out the diabatic picture, electronic couplings between these diabatic states are also needed, describing how population flows between the diabats as the system evolves. This is in contrast with the adiabatic (Born-Oppenheimer) picture, where electronic states are always taken to be eigenstates of the electronic Hamiltonian with no direct coupling to each other, no matter the geometry of the system. Nonetheless, the diabatic picture proves itself quite useful, producing PESs for dynamics that vary slowly with nuclear coordinates and thus avoid sharp changes where errors can accumulate. Addi- tionally, diabats are invoked to assign vibronic transitions in spectroscopy and for qualitative descriptions of molecular bonding (as for the LiF example of Figure 42), electron transfer (cf section 3), and proton tunneling. However, since the diabatic states are not eigenstates of the electronic Hamiltonian, chemistry (that is, reactions) in the diabatic basis must necessarily incorporate multiple diabats, and the diabatic coupling between them:
En savoir plus

173 En savoir plus

HOPMA: Boosting protein functional dynamics with colored contact maps

HOPMA: Boosting protein functional dynamics with colored contact maps

cutoff. There are multiple ways to compute protein transitions with NMA, for example, using normal modes in internal coordinates ( 77 – 81 ). Here, we used the NOnLinear rigid Block (NOLB) NMA, a Cartesian nonlinear approach that extrapolates the instantaneous eigenmotions as a series of twists ( 82 ). NOLB is computationally very efficient, as it can be applied to systems as large as viral capsids or ribosomes at all-atom resolution on a personal laptop ( 82 ). It produces conformations with high stereochemical quality and is able to predict motions with different degrees of collectivity, from localized and disruptive motions to highly collectives ones ( 68 ). We place ourself in the context where we know the target structure and we use this knowledge to determine the amplitudes of the initial struc- ture’s normal modes. This set up allows obtaining the optimal (or close-to-optimal) transitions within our framework. In practical applications, when the target is not know, the sam- pling may be guided by additional information ( 83 , 84 ), such as experimental small-angle scattering profiles ( 25 – 27 , 85 , 86 ), low-resolution electron density maps ( 15 – 22 ), intensities from crystallographic scattering experiments ( 87 , 88 ), complemen- tarity of the molecular shapes upon binding ( 28 – 36 ), or other type of data such as contacts inferred from coevolutionary signals extracted from protein sequences ( 89 – 91 ).
En savoir plus

11 En savoir plus

Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates

Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates

Figure 21: Supersonic case §5.3 iso-Mach number lines (left) ∆ M = 1.5, iso-lines of first-component of lift adjoint vector (right) In the supersonic case (Figures 18 to 20), a detached shock-wave is formed ahead of the body, and the supersonic flow is uniformly constant upstream of it. Thus, the sole upstream influence is concentrated in the shock geometry information, a zone of evident mesh concentration requirement. In particular, the fine mesh requirement of the stagnation-point streamline disappears upstream the shock. The subsonic bubble about the stagnation point and a small zone downstream close to the profile, upper and lower side, are regions of very high mesh density. The secondary shock is a fish-tail shock-wave; its base is at the trailing edge. Here, the region downstream the profile is supersonic and has no upstream influence. More precisely, the adjoint fields of the lift (first component figure 21), describes the geometric zone of flow influence for the lift (and actually the other near field functions functions) : a large angular section based on the trailing edge has no influence on lift (and other near field function values). Conversely,
En savoir plus

25 En savoir plus

Brain Transfer: Spectral Analysis of Cortical Surfaces and Functional Maps

Brain Transfer: Spectral Analysis of Cortical Surfaces and Functional Maps

Shape Variability – We reuse the harmonic weights computed in the former exper- iment on cortical alignment. PCA is applied on all 16 transferred harmonic weights, each representing surface point coordinates. The target space is chosen arbitrarily as the harmonic basis of the first subject. Our aim is to explore the variability of weights in a common space, regardless of finding an optimal unbiased space. Fig. 10-a shows the reconstruction of cortical surfaces using the average of transferred harmonic weights, and the first 3 principal modes of variations within 2 standard deviations. The first mode appears to capture a global rotation of the cortices, which inter- estingly, were not initially aligned in space. This indicates that Brain Transfer can correctly handle datasets that are not initially aligned. Further modes capture vari- abilities in the cortical folds, such as, shape differences in the occipital and temporal lobes.
En savoir plus

13 En savoir plus

Existence of a density functional for an intrinsic state

Existence of a density functional for an intrinsic state

A more general argument is possible, with more than one kind of identical particles. Trial states can be, for instance, products of determinants, one for each kind of fermions, and permanents, one for each kind of bosons. Consider for instance the ammonia molecule, with i) its active electrons, ii) its three protons and iii) its nitrogen ion. It is trivial to include a center-of-mass trap into H to factorize into a spherical wave packet the center of mass motion of this self-bound system and avoid translational degeneracy problems. The complete Hamiltonian H, trial states φ and density operators B depend on and describe simultaneously the electron, proton and nitrogen ion coordinates and momenta. A DF in just the electronic density space, however, results from the definition,
En savoir plus

5 En savoir plus

Fine-grain atlases of functional modes for fMRI analysis

Fine-grain atlases of functional modes for fMRI analysis

p ~ 200,000 voxels GLM on DiFuMo reduction, p = 1024 modes Group-level z-map: complex sentence Figure 3: Overlap between GLM maps obtained with func- tional atlases and voxel-level analysis. Top: The overlap is measured with the Dice similarity coefficient. The black line gives a baseline the mean overlap between voxel-level contrast maps over several random selections of sessions per subject. The figure gives Dice similarity scores between the GLM maps computed with sig- nals extracted on functional atlases and at the voxel-level, after re- construction of full z-maps and voxel-level thresholding with FDR control. The best similarity is achieved for highest dimensional- ity, though 1024-dimensional DiFuMo atlas largely dominates 1000- dimensional Schaefer parcellation and 1095 MIST. Each point is the mean and the error bar denotes the standard deviation over contrast maps. Bottom: The activity maps encoded on 1024-dimensional space capture the same information as voxel-level analysis, while be- ing smoother. Figure A7 shows activity maps for all atlases.
En savoir plus

26 En savoir plus

Fine-grain atlases of functional modes for fMRI analysis

Fine-grain atlases of functional modes for fMRI analysis

maps, using the Dice ( 1945 ) similarity coefficient. We also perform an intra-subject analysis detailed in AppendixD . Data. We consider the Rapid-Serial-Visual-Presentation (RSVP) language task of Individual Brain Charting (IBC) (see Pinho et al. , 2018 , for experimental protocol and pre- processing). We model six experimental conditions: com- plex meaningful sentences, simple meaningful sentences, jabberwocky, list of words, lists of pseudowords, conso- nant strings. β-maps are estimated for each subject using a fixed-effect model over 3 out of the 6 subject’s sessions. We randomly select 3 sessions 10 times to estimate the variance of the Dice index across sessions. As a baseline, we evaluate the mean and variance of the Dice index across z-maps when varying the sessions used in voxel-level GLM. 3.3. Decoding experimental stimuli from brain responses
En savoir plus

22 En savoir plus

Density-functional theory, finite-temperature classical maps, and their implications for foundational studies of quantum systems

Density-functional theory, finite-temperature classical maps, and their implications for foundational studies of quantum systems

2. Density functionals and Bohmian mechanics To be specific, we consider a system of electrons subject to an external potential v ext (r), and interacting with one another via the Coulomb potential. In many cases this external potential is produced by a fixed set of nuclei, as in molecular physics or quantum chemistry calculations at T = 0. However, the ions cannot be treated as fixed in dealing with liquid metals and hot plasmas. Hence, in discussing true hybrid systems, we allow the nuclei (or ions) to be a second interacting subsystem, interacting with the electron subsystem, and the finite-T discussion becomes not only appropriate, but also necessary. However, in simplified models, e.g., the jellium model, the ions provide a fixed uniform positive distribution of charge that exactly neutralizes the negative charge of the electron distribution. The q → 0 singularities in the Coulomb potential 4π/q 2
En savoir plus

9 En savoir plus

High-throughput free energies and water maps for drug discovery through molecular density functional theory

High-throughput free energies and water maps for drug discovery through molecular density functional theory

HAL Id: cea-02340016 https://hal-cea.archives-ouvertes.fr/cea-02340016 Submitted on 30 Oct 2019 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

2 En savoir plus

Constrained Density-Functional Theory--Configuration Interaction

Constrained Density-Functional Theory--Configuration Interaction

with a promolecule density approach. 9 We will use this approach again to obtain practical constraint values for reactant and product. 2.2.3 Promolecules CDFT is designed to construct electronic states of fixed character at arbitrary geometries, even those where fragments overlap, but sometimes it does not perform as well as might be hoped in such close-contact geometries. One of the sorest impediments to its ability to do so is the choice of atomic population prescription. In cases of close approach, the real-space constraint potentials must distinguish between fragments in regions where the density is nonzero, so that assignment of density in that transition region to fragment “A” or fragment “B” is ambiguous, a clear weakness of the available charge prescriptions. The simplest example that shows this ambiguity is H + 2 , with a single electron and two protons. Formally, we can constrain the electron to lie only on one proton (“A”), but when the two protons begin to approach each other, any real-space-based atomic population scheme will begin to assign some fraction of this electron to the second nucleus (“B”), for any physically reasonable density corresponding to a constrained H − H + configuration. Thus, the formal charge constraint putting a full electron on atom A is unattainable with present charge prescriptions, and the numerical value of the constraint must be adjusted to accommodate their failings.
En savoir plus

136 En savoir plus

Matched coordinates for the analysis of 1D gratings

Matched coordinates for the analysis of 1D gratings

lines, hence the name matched coordinates. Unfortu- nately, in this formulation, the coefficients of the differ- ential system of Maxwell’s equations are not constant which means that the solution in the grating region cannot be obtained thanks to a modal method. Instead, in a manner similar to what was done in [9], we intro- duce a pseudo-spectral approach to solve the longitu- dinal dependence of the field along the grooves. In the case of deep gratings the modulated region may be di- vided into thin enough layers in order to work with smaller matrices. Accurate and fast electromagnetic simulation methods gained considerable importance with the rise of modern technology. Particularly semi- conductor manufacturing and metrology play a sig- nificant role in pushing the limits. Famous examples are optical scatterometry, a.k.a. as optical CD (OCD) where electromagnetic modeling paved the way for the fast and non-destructive measurement of techni- cal structures in the nano-meter range [10]. However, OCD is just the tip of iceberg. Model based metrology methods are quickly evolving everywhere in dimen- sional metrology [11]. Other examples are laser focus scanning and white light interferometry [12]. Many technical surfaces either consist of metals or are coated with it. In addition, the profiles are steep or even over- hanging. While the RCWA exhibits convergence is- sues for metallic materials both related to the required stair-case approximation of the profile as well as to the Gibb’s phenomenon related to the Fourier expan- sion, the C-method struggles with multiple materials and very deep profiles. Here, the matched coordinates method may bring relief. We will demonstrate this with some practical examples.
En savoir plus

12 En savoir plus

On the Stability of Functional Maps and Shape Difference Operators

On the Stability of Functional Maps and Shape Difference Operators

W X ←− − ˜ K X (i, j). Using these constructions, we observe that the robustness is evi- denced in the results from our PCD setting as well. In Figure 10 , we tested with point clouds sampled from horses (8431 points) and cats (7207 points). Point clouds X ,Y come from two meshes in point- to-point correspondence, the highlighted functions on Y are shown to be consistent with the ones computed in the mesh setting on N (see the right-most column). We then resampled point clouds on shape Y while keeping the total number of points unchanged, and computed the highlighted function with respect to X and the resam- pled point cloud. At the end, a further step was taken to add noisy points on the resampled point clouds. Given a point cloud, we first randomly select n p points from it. Then for a selected point p, we perturb p = (p x , p y , p z ) ∈ R 3 to (p x + dx, p y + dy, p z + dz) where dx, dy, dz are one-dimension random variables distributed normally with mean 0, and standard deviation ¯ d , which is the median of the distances from a point to its nearest neighbor. Repeating the
En savoir plus

13 En savoir plus

Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

(2) instead. Here, q 1 and q 2 describe the charges of the par- ticles while a is the usual softening parameter (atomic units e = m = ¯ h = 1 are used throughout this paper). We use a = 1 for all our calculations. Mathematically, it is straightforward to show that the Hamiltonian (1) is equivalent to a single particle in N dimensions, moving in an external potential consisting of all the contributions from v ext and v int . The corresponding Schr¨odinger equa- tion can, hence, be solved by any code which is able to treat non-interacting particles in the correct number of dimensions in an arbitrary external potential. Due to the Hamiltonian being symmetric under particle interchange, x j ↔ x k , the solutions of the Schr¨odinger equation can be chosen as symmetric or antisymmetric under such an exchange. For the simplest case of two interacting elec- trons both the symmetric and antisymmetric solutions are valid, corresponding to the singlet and triplet spin configurations, respectively. However, for more than two electrons one needs to separately ensure that the spatial wave function is a solution to the N -electron problem.
En savoir plus

5 En savoir plus

Kernel density estimation and transition maps of Moldavian Neolithic and Eneolithic settlement

Kernel density estimation and transition maps of Moldavian Neolithic and Eneolithic settlement

which is de fined in the same way as the Normalised Difference Vegetation Index, i.e.: NDR ¼ ðT1þT0Þ ðT1−T0Þ When set against salt resources this cartographic formula allows us to visualise and compare the settlement dynamic between two chronological sequences and thus to distinguish between the areas that are most attractive (from 0.33 to 1), most stable (from − 0.33 to 0.33) and those that have fallen into decline/disuse (from − 1 to − 0.33).

8 En savoir plus

Correlation maps: A compressed access method for exploiting soft functional dependencies

Correlation maps: A compressed access method for exploiting soft functional dependencies

space and index lookups during query processing consume fewer I /O operations. However, an even more significant improvement comes from reduced index maintenance costs. Indexes have to be kept up-to-date as the underlying data change through inserts or deletes. Accessing the disk on every update is prohibitively expensive. Thus, the universally applied solution is to keep modified pages in memory and delay writing to disk for as long as possible. Unfortunately, RAM is a much more limited resource than disk space and only a small fraction of a typical B+Tree can be cached. As we show in Experiment 3 of Section 7.2, maintaining as few as 5 or 10 B +Trees can lead to a dramatic slowdown in overall system performance. CMs on the other hand can be usually be fully cached in memory as they are quite small; this leads to substantially lower update cost. This means that it is realistic to maintain a large number of CMs, whereas it may not be practical to maintain many conventional B+Trees. In the rest of this section, we describe how CMs are structured, built, and used.
En savoir plus

13 En savoir plus

Sensitivity analysis for the Euler equations in Lagrangian coordinates

Sensitivity analysis for the Euler equations in Lagrangian coordinates

The Godunov method. In this paragraph, we present the usual Godunov method based on the exact resolu- tion of the Riemann problem including the source term, and associated with the ini- tial data U(x, 0) = U L 1 (x<x c ) + U R 1 (x>x c ) and U a (x, 0) = U a,L 1 (x<x c ) + U a,R 1 (x>x c ) . The details are not reported here but one has been able to prove that the analytical solution is known and that its structure is resumed in Figures 1 - 2 . In particular, the solution for the state consists of two waves, which can be either shocks or rarefaction waves, and whose speed can be computed exactly. On the other hand, the sensitiv- ity has the same two-wave structure, however both of the waves are discontinuities. This simplification for the sensitivity is due to the fact that we are considering a reduced Euler system, under barotropic conditions (cf. [ 6 ]).
En savoir plus

9 En savoir plus

The classical-map hyper-netted-chain (chnc) method and associated novel density-functional techniques for warm dense matter

The classical-map hyper-netted-chain (chnc) method and associated novel density-functional techniques for warm dense matter

e −(H e / T e +H i / T i +H ei / T ei ) (22) Here the total Hamiltonian H is rewritten in terms of H e , H i , and the electron-ion interaction H ei which is again a Coulomb potential. We have included a cross-subsystem temperature T ei which is simply T for equilibrium systems. If a Born-Oppenheimer approximation is used, the electrons “do not know” the temperature of the ions, and vice versa. For equi- librium systems, a Born-Oppenheimer correction can be introduced, e.g., as in Morales et al. [45]. How- ever, the “add on” correction introduced by Morales et al. will change the virial compressibility, leaving the small k → 0 behavior of the proton-proton struc- ture factor unaffected, and hence the effect on the compressibility sum rules has to be examined. In any case the DFT implementations in codes like VASP cannot deal correctly with the case T i = T e , and it
En savoir plus

13 En savoir plus

Double-hybrid density-functional theory made rigorous

Double-hybrid density-functional theory made rigorous

much less accurate functional than BLYP for atomiza- tion energies. Unfortunately, there is no short-range exchange-correlation functional based on BLYP available yet. In fact, it is a practical advantage of the double hybrids without range separation that they do not re- quire development of new density functional approxima- tions. The range-separated coupled-cluster calculations on an extension of the G2/97 set of atomization ener- gies by Goll et al. [67] show that a better accuracy can be reached by using a short-range exchange-correlation functional based on the TPSS functional [76]. For the reaction barrier heights of Table IV, 1DH-BLYP is on average more accurate than B2-PLYP, with a smaller MAE (1.4 vs. 2.0 kcal/mol) and an even smaller ME (-0.8 vs. -1.8 kcal/mol). The range-separated double hybrid RSH-PBE(GWS)+lrMP2 performs about equally well for the barrier heights, with a MAE of 2.1 kcal/mol and a ME of 1.1 kcal/mol. Notice that we have used the aug-cc-pVQZ basis for this DBH24/08 set. Diffuse basis functions are indeed important for describing the charged species in the nucleophilic substitutions, particularly for 1DH-BLYP and B2-PLYP and to a lesser extent for RSH- PBE(GWS)+lrMP2. For comparison, the MAEs of these three methods with the cc-pVQZ basis are 2.1, 2.7, and 2.4 kcal/mol, respectively.
En savoir plus

11 En savoir plus

Open problems in nuclear density functional theory

Open problems in nuclear density functional theory

ρ 0 (r). The final result is a scalar functional, F 0 [ρ 0 ]. Hence an open problem : for such monopolar profiles ρ 0 , what is the signature for deformation? It is likely that the slope of ρ 0 at the nuclear surface will be weaker for deformed nuclei than for spherical ones, but precise criteria distinguishing hard from soft nuclei are desirable, not to mention criteria identifying triaxial nuclei, halo nuclei, etc. Can one use properties of moments of the profiles? Such a theory with radial profiles, in one dimension rather than the three dimensions of usual nuclear (E)DF theories, provides a considerable simplification of numerics, but demands somewhat subtle criteria for the identification of deformations.
En savoir plus

11 En savoir plus

Show all 10000 documents...