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:

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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 ).

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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,

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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.

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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,

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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.

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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

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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.

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.

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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.

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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**

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(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.

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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).

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.

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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

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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.

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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.

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