Hybrids in Momentum Space

For the following considerations, the orientations of the hybrids are not relevant, and we therefore mix only pz orbitals with s-functions, resulting in an orbital that points along the polar axis 2 in both spaces. This takes the following form for the n = 2 shell  

In contrast, the real part of the momentum hybrid ipasp has a spherical node (at p = J(p2) = Z/2), whereas the imaginary part has a planar one (at 6 = 7t/2). This means 

It is widely known that total momentum densities for atoms are not always monoton-ically decreasing [10], In fact the degree of non-monotonicity is dependent on the degree of p-population in an atom. This fact is visible as well in the shape of spa hybrids in momentum space. 

In Fig.(l), we display the momentum density contributions of commonly encountered hybrid orbitals, obtained from hydrogenic eigenfunctions with Z = 1. The figure shows surface plots of the densities for in the 2-plane for a = 1, 2 and 3. It may be seen that, while the sp hybrid exhibits a maximum at p = 0, greater p-contributions flatten this maximum out, leading to a plateau for sp2, and finally a saddle point for sp3. 

All of these densities feature two points in the zz-plane where the density vanishes exactly. They are situated on the x-axis, as sections along that axis demonstrate clearly. We show those in Fig. (2). Independently of the mixing coefficient a, those nodal points occur at x = 1/2 on each equatorial axis. They are the intersection of the aforementioned nodal circle with the displayed plane. 

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The two concepts have on occasion been brought together Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO s). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6], We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do... 

Figure 1. Surface plot of the orbital densities for spa hybrids in momentum space. The hybrids are based on the hydrogenic wave functions. The three plots pertain to a = 1, 2 and 3, respectively. The hybrids point in the 2-direction. A section through the density in the 22-plane is displayed.

The momentum-space expression was first given by Podolsky and Pauling [3] in 1929. Note, that for any real position function Rh, the corresponding momentum radial function Rh will be either purely real or purely imaginary, depending on whether the angular part of the orbital is even or odd (see also [8]). The factor (—O >n Eqn.(5) has, e.g., the consequence that s-type and p-type functions do not mix in momentum space, which leads to hybrids that have a different nodal structure. 

Figure 4. The sp, sp2, and sp3 hybrid density functions in the 12-plane of position space. As is often the case, orbitals that are quite different from one another in momentum space, can appear very similar in the corresponding position space representation.

The second class of spri-type hybrids that we will treat here are situated in the equatorial plane, i.e. at 6 = 7t/2. They consist of a linear combination of s, px and dx2 y2 orbitals, and have proven useful in describing the bonding in square-planar complexes. Their form (in momentum space) is ... 

It is a simple matter to derive expressions for the moments of the hybrid orbital densities. In momentum space, the expressions will take the form,... 

Among the more interesting qualities of momentum space hybrids, is the lack of strong directional asymmetry, this being one of the most noticeable characteristics of position space hybrids. Indeed, it is this directional asymmetry which has made hybrid orbitals so useful for describing directional bonds. In momentum space, however, the hybrids are inversion-symmetric and this is shown to have a profound effect on the nature of these orbitals. 

Another difference is the nodal structure of these atomic contributions to the total density. The hybrid orbitals as we know them, in position space, exhibit nodal surfaces, i.e. two-dimensional subspaces on which the density vanishes. This dimensionality is reduced in momentum space. Here, the nodes are invariably one dimensional, i.e. curves that are formed by the intersection of real and imaginary nodal planes. 

Finally, (for atoms), the momentum densities corresponding to hybrid orbitals exhibit a few basic extremal features close to the origin. These depend on the weight that is given to s, p and d contributions, and they determine the basic look of the density. Outwardly, momentum-space hybrids share one feature with a related experimental quantity, the Compton profile they all look alike. On closer inspection, however, there are a variety of complex features, mainly arising from the nodal structure of the orbitals. Apart from the obvious use of hybrids in position space for the description of bond situations, there is another feature that has always captured the interest of scientists and laymen their intricate structure. This feature is less apparent in momentum space, but it is still present. If nothing else, its enjoyment makes a close look at these entities worthwhile. 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. 

Figure 5. Nodal surfaces of a sp d2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the -axis. The intersection of the two types of nodes are two circles around the -axis. The right-hand plot displays a cut through the a -plane. Note that the (polar) -axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.

For all momentum densities, the origin of momentum space is necessarily a critical point. This arises from the inversion center and the requirement of continuity. However, the topology at that point may vary considerably for the hybrids considered here. We pointed out already in the previous section that the sp -hybrids do not always exhibit a pair of off-center maxima. For hybrids containing d-functions, the picture is further complicated. 

Moments of the spa hybrids in both coordinate and momentum spaces. 

Complete expressions for the hybrid charge density moments, in both position and momentum space, are given in Tables 1 and 2. The former table contains results for the spa hybrids while the latter gives results for the spadb hybrids. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

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