Gaussian Wave
While a Gaussian wave packet and a Thouless single-determinantal state are different types of functions, both nevertheless belong to a vast category of functions named coherent states (CS).
From: Advances in Quantum Chemistry, 2013
Related terms:
Time-Dependent Treatment of Molecular Processes
Yngve Öhrn, in Advances in Quantum Chemistry, 2015
4.1 Gaussian Wave Packet as a Coherent State
A Gaussian wave packet in one dimension can be expressed as (ℏ=1)
where we take the view that the parameters p and q are time dependent, while the width parameter b is time independent. The interpretation of these parameters are evident from the definition of quantum mechanical averages, i.e.,
Since the average value x−q=0, it follows that q=x, which is the average position of the wave packet. The average momentum of the wave packet
defines the parameter p. The square of the width of the wave packet
making the width Δx=x−q21/2=b/2.
The Gaussian wave packet has a number of interesting properties. For instance, it has a minimal uncertainty product ΔxΔp = 1/2 in units of ℏ. This follows from
and Δp=p^2−p21/2=1/b2, (ℏ=1).
In addition, the Gaussian wave packet can be written as a displaced harmonic oscillator ground state, such that
i.e., the oscillator ground state is displaced, x→x − q, and boosted 0 → p. This can be seen from
The Gaussian wave packet is a coherent state and can be expressed as a superposition of oscillator states. This means that
where n is a harmonic oscillator eigenstate, a and a† are harmonic oscillator field operators, and z is a suitable complex combination of wave function parameters.
This can be seen from the result
where, since x and p^ do not commute, the last step is nontrivial. Introducing the complex parameter z=(q/b+ibp)/2 and observing that the harmonic oscillator field operators can be expressed as
we can write za†−z*a=−iqp^−px and
The last expression is the “classical” or canonical coherent statez. The Baker–Campbell–Hausdorff (BCH) formula yields
which is true for the case when the commutator A,B− commutes with A and B. When applied to
the BCH formula yields
The Gaussian wave packet in this form is the original “coherent state.” Generalizations of this concept have been made, in particular the work of Perelomov13 has introduced so-called group-related coherent states. Such a state is formed by the action of a Lie group operator exp∑mzmFm acting on a reference state |0〉, where {zm} are the, in general complex, Lie group parameters and Fm are the generators of the corresponding Lie algebra. The reference state is usually a lowest weight state and called the fiducial state. It is commonly invariant to some of the group elements, thus defining a so-called stability group of the fiducial state. The parameters labeling the coherent state are then associated with the left coset of the Lie group with respect to the stability group. This assures nonredundancy of parameters. The canonical coherent state has this form in terms of the so-called Weyl group, whose Lie algebra generators are 1,a,a†. The one parameter stability group is just the phase factor eiα and the coset representative is eza†−z*a.
The scalar product of two coherent states
i.e., a nowhere vanishing continuous function of the parameters. The canonical coherent state has the property that
which can easily be seen from
which means that e−za†aeza†0=z0. Furthermore, the coherent state for all the values of the complex parameter z is a set of states satisfying the resolution of the identity
This result permits us to write
which illustrates the overcompleteness of the set z. Thus, as a set of functions labeled by the continuous complex parameter z, the coherent state satisfies the resolution of the identity and is inherently linearly dependent.
Considering the time evolution of a harmonic oscillator with z as the initial state, we obtain
This shows that the coherent state evolves into other coherent states by a time-dependent label change that follows the classical oscillator solution.
Application of the TDVP to the wave-packet dynamics with the coherent state z is straightforward. We note that
and that the dynamical equations become
since C=∂2lnS/∂z*∂z′∣z′=z=1. The equation iŻ=ωz becomes in more detail
assuming a constant width wave packet. One easily deduces that
i.e., in an oscillator field with b=1/mω the Gaussian wave packet is coherent and that its average position has a harmonic motion q(t)=q0cosωt+p0mωsinωt, while pt=p0cosωt−mωsinωt.
Analysis of transverse modes of laser radiation
In Computer Design of Diffractive Optics, 2013
The mode properties of the generated beams
We show further that the light fields formed in this manner do indeed have modal properties.
When illuminating a phase DOE (8.128) with a flat or Gaussian wave a light field fomrms in the spectral plane with the complex amplitude which is close to the given mode. Another Fourier transform of the resulting complex amplitude distribution should form a light field proportional to the illuminating beam. However, the introduction of the diaphragm to the spectral plane, as in [51], produces in the image plane a light field which is also close to the given mode. Figure 8.32 shows the optical scheme for the formation of Gaussian modes.
Fig. 8.32. Optical scheme for the formation of Gaussian modes.
The light beam from a helium-neon laser 1 is collimated by 2 and illuminates the DOE 4, whose phase is proportional to the sign function of the corresponding polynomial. Diaphragm 3 is adjusted to the optimum size for the formed mode [− a, a] [47]. Spherical lens 5 forms a spatial spectrum at the focal distance from which the diaphragm 6 separates an effective part [− b, b]. The image obtained using spherical lens 7 in the plane 8 has a complex amplitude, showing the modal character of the generated field.
Figure 8.33 shows the formation of the fifth GH mode. In this case, for clarity, all the distributions of amplitude and intensity are aligned with respect to the maximum value, not its energy characteristics. Figure 8.33a shows the distribution of the amplitude of the ideal mode (line 1) and binary phase DOE 4 (line 2). Figure 8.33b shows the intensity distribution obtained in the spectral plane in illumination of the DOE with a flat beam (line 2) and for comparison the intensity distribution of the ideal mode (line 1). The position of the diaphragm 6 is indicated by the dotted line. Figure 8.33c shows the intensity distribution of the image plane 8 (line 3), the ideal mode (line 1) and one of the Fourier transforms of the received image (line 2). As shown by comparison of Fig. 8.33b and 8.33c, the intensity distribution after the third Fourier cascade (Fig. 8.33c, line 2) is closer to the ideal mode than after the first Fourier cascade (Fig. 8.33b, line 2). Moreover, it has been obtained without the introduction of the diaphragm in the image plane 8.
Fig. 8.33. (a – d) Formation of the fifth mode GH using a binary phase DOE.
The modal nature of the generated beam is also confirmed by the distribution of the appropriate phases (see Fig. 8.33d). When comparing the phase distributions it must be taken into account that after each of Fourier transform the modes GH acquire a phase shift of πn/2, where n is the number of modes. Table 8.8 shows: the phase shift (taken with respect to modulus 2π) in the image plane – φ1 and the phase shift in the following spectral plane – φSS. It is clear that for the GH modes whose number n is a multiple of 4, the phase portrait will be the same in both the image plane and in the spectral plane. For even modes GH, but not multiples of 4, complex distributions in the image plane and in the spectral plane will be in opposite directions.
Table 8.8. Characteristics of the formation of GH binary modes of DOE
| 1 | 2.25 | 10.09 | 85.59 | 3.50 | 12.00 | 85.45 | 6.09 | π | |
| 2 | 2.70 | 14.91 | 83.33 | 3.32 | 7.98 | 83.22 | 12.47 | 0 | π |
| 3 | 3.10 | 15.51 | 81.76 | 3.42 | 6.67 | 81.49 | 13.82 | π | π/2 |
| 4 | 3.42 | 16.70 | 80.50 | 3.57 | 9.32 | 80.03 | 15.20 | 0 | 0 |
| 5 | 3.75 | 18.60 | 79.45 | 3.75 | 12.48 | 78.71 | 15.92 | π |
Figure 8.34 shows graphs of the standard deviation of the intensity distribution from the ideal in the spectral plane δS for the fourth (line 1) and fifth (line 2) modes GH, as well as in the image plane δ1: for the fourth (line 3) and 5-th (line 4) modes, depending on the size of the diaphragm 3 [− a, a]. It is interesting to note that if the global minima in Fig. 8.34 (lines 1, 2) correspond to the optimal size of the DOE, the additional (local) minima coincide with the last zeros of Hermite polynomials.
Fig. 8.34. Graph of the standard deviation of the distribution of intensity from the ideal distribution in the spectral plane δS for the fourth (line 1) and fifth (line 2) modes GH in relation to the size of the DOE [− a, a], as well as in the image plane δI for the fourth (line 3) and fifth (line 4) modes, depending on the size of the aperture [− b, b].
The optimum size of the DOE [− a, a] and the diaphragm 6 in the spectral plane [− b, b] are given in the summary table 8.9. The table shows that the standard deviation of the intensity distribution in the image plane from the ideal distribution, δI, are as a rule smaller than the deviations in the spectral plane δS. The deviation in the following spectral plane δss (i.e. the Fourier transform of the image) is smaller than δS. This means that the optical system in Fig. 8.32, repeated consecutively a few times, is an approximation of the cavity.
Table 8.9. Summary results for the two-dimensional case
| rectxarectxb | 2.2 1.7 |
2.2 2.2 |
no | no | 28.8 14.4 |
63.8 72.5 |
|
| (1.0) | x2a2+y2b2=1 | 2.2 1.85 |
2.2 2.45 |
no | no | 26.9 14.3 |
70.1 74.6 |
| exp−x2a2exp−y2b2 | 1.9 1.9 1.9 1.5 1.5 1.5 |
1.9 1.9 1.9 2.05 2.05 2.05 |
no 2.7r 3.0e no 2.7r 3.0e |
no 3.1r 3.55e no 3.1r 3.55e |
28.5 22.5 24.5 18.3 15.4 15.7 |
75.2 74.9 75.0 80.5 80.3 80.4 |
|
| rectxarectxb | 2.25 | 2.25 | no | no | 14.4 | 72.3 | |
| (1.1) | x2a2+y2b2=1 | 2.64 | 2.64 | no | no | 15.6 | 69.5 |
| exp−x2a2exp−y2b2 | 2.1 2.1 2.1 |
2.1 2.1 2.1 |
no 3.15r 3.55e |
no 3.15r 3.55e |
26.8 20.3 21.0 |
65.1 65.6 65.7 |
|
| rectxarectxb | 2.5 2.6 |
2.5 2.2 |
no | no | 30.6 18.3 |
68.5 71.3 |
|
| (1.2) | x2a2+y2b2=1 | 3.0 3.2 |
3.0 2.6 |
no | no | 31.8 23.5 |
65.4 67.3 |
| exp−x2a2exp−y2b2 | 2.5 2.5 2.7 2.7 |
2.5 2.5 2.2 2.2 |
no 3.4r no 3.4r |
no 3.05r no 3.05r |
49.1 39.4 45.2 37.15 |
61.5 61.5 63.1 62.3 |
Figure 8.35 shows the plots of the diffraction efficiency ε (8.136) (lines 1 and 2) and the standard deviation of the intensity distribution from the ideal distribution (8.135) (lines 3 and 4) depending on the number of Fourier transforms, k, for the fourth (line 1, 3) and fifth (lines 2, 4) modes GH. In this case, k = 0 corresponds to the plane of the DOE, k = 1 to the first spectral plane (see δS, εS in Table 8.8), k = 2 is the first image plane (see δS εs: in Table 8.8), k = 3 is the second spectral plane (see δss in Table 2.4), k = 4 is the second image plane.
Fig. 8.35. Graphs of energy efficiency ε (lines 1 and 2) and standard deviation of the intensity distribution from the ideal distribution δ (lines 3 and 4) depending on the number of Fourier transforms, k, for the fourth (line 1, 3) and fifth (lines 2, 4) modes GH.
Figure 8.35 shows that after the stage of low-pass filtering in the first spectral plane (k = 1) further energy losses are negligible. This means that the entire energy picture, as well as the spectra, is concentrated on a finite interval of spatial frequencies. This is also characteristic of the Gaussian modes.
Advances in Additive Manufacturing and Tooling
S. Kumar, in Comprehensive Materials Processing, 2014
10.05.3.2.3 Laser Spot Size
Spot size could be defined as a laser beam diameter on powder bed. For Gaussian wave, the diameter lies at e−2 of the peak intensity of the beam. Its size from 30 to 600 μm is used in the process.
Variable spot size is required to achieve variable laser energy density, precision, and speed of production. Laser energy density required for processing depends upon the spot size. The size could be increased or decreased to get low or high energy density, respectively. The minimum spot size depends upon the wavelength and quality of the laser beam.
For making small features, thin walls, or small holes and getting high resolution of the product, small spot size is needed. However, processing the powder bed with small spot size takes longer duration to complete and increase the production time. Therefore, an optimum spot size is searched that does not compromise with the accuracy of the product and at the same time does not take unreasonable long production time. It could also be achieved by using variable spot sizes. For processing the bulk, larger spot size could be used while for making boundaries or thin features, small spot size could be used.
Gravity Waves
Nikolaos D. Katopodes, in Free-Surface Flow:, 2019
3.5.4 Dispersion of a Composite Wave
For a progressive wave that is composed of several sinusoids, the Fourier integral representation reads
where ηˆ(k) is the Fourier transform of the wave. Since there are multiple wave numbers involved in this wave, we may write k=k0+κ, where k0 represents the “central” or “carrier” wave number of the wave, and κ is allowed to vary over the entire range of possible wave numbers. Then, Eq. (3.56) can be written as follows
The frequency term in the exponential inside the integral can be approximated by a Taylor series, as follows
Thus, if we retain only the first-order term, the exponential can be written as follows
where c0=ω0/k0 is the phase speed of the carrier wave, and
is called the group velocity. It represents the speed with which the entire wave packet is traveling. Therefore, if we pass the carrier wave part though the integral sign, we obtain
or, more compactly,
where F0 is a functional representation of the envelope of the wave packet's amplitude, which travels with speed Cg.
Similarly, we can describe the propagation of the wave by including the second-order term in the Taylor expansion, which yields
This shows a different behavior of the wave. However, before we can assess the physical meaning of the changes, we need to narrow out analysis to a specific wave form.
3.5.4.1 Gaussian Packet
The propagation of the individual components and the wave packet becomes transparent when we consider a specific composite wave, for example, a Gaussian wave packet. Let us assume that the wave envelope is described by the density function given by Eq. (I-1.86) that was introduced in section I-3.6.2 in connection with the process of molecular diffusion. For present purposes we can write the Gaussian density function as follows
Hence, the distribution is centered at point x0, and has a spatial variance σx2. Let us first differentiate the distribution (3.63), as follows
Then, taking the Fourier transform of both sides, we obtain
Therefore, after rearranging and integrating, we arrive at the following expression
or
Finally, we can write
where σk=σx−1. Therefore, the Fourier transform of the Gaussian density function is also Gaussian. However, its width is the inverse of the width of the corresponding physical Gaussian. Notice also that the spatial shift, x0, corresponds to a rotation, k0, in Fourier space.
In summary, the Gaussian density function, (3.63), contains a set of wave numbers clustered around the carrier wave number, k0. For a uniform distribution, σx→∞, thus k→0. Conversely, infinitely many wave numbers are needed to describe a sharp Gaussian, i.e. as σx→0. This offers a new interpretation for the Gaussian density function as a wave packet.
OPTICAL COHERENCE TOMOGRAPHY THEORY
Mark E. Brezinski MD, PhD, in Optical Coherence Tomography, 2006
5.4.3 Relationship between the Paraboloidal and Gaussian Wave
Before moving on to analyzing the detailed form ofEq. 5.105, for a Gaussian function, the relationship between the complex envelope of a paraboloidal and Gaussian wave shall be expanded on. The complex envelope of a paraboloidal wave is given by the general formula:
where exp{−ik(x2+y2)/2(z)} modulates the envelope. For a Gaussian function, we replace z by q(z) where q(z) = z + iz0 and z0 is a constant. The equation modulating our complex envelope is now:
The shift by iz0, we will see, has very significant implications.
Computational Modelling of Nanoparticles
Lucas Garcia Verga, Chris-Kriton Skylaris, in Frontiers of Nanoscience, 2018
8.1.3 Basis sets
Kohn–Sham DFT reintroduces the idea of noninteracting particles wavefunctions, which need to be calculated with computationally efficient approaches that accurately represent these complicated mathematical functions. Two widely established and used classes of basis set on DFT calculations are the Gaussian and plane-wave basis set.
Gaussian basis set, and other atom centered functions, such as Slater-type orbitals (STOs), are constructed under the local combination of atomic orbitals (LCAO) ansatz, where the molecular orbitals ψi are expanded in terms of linearly combined basis functions θα with expansion coefficients Cαi.
Under the LCAO approach, using STOs as basis functions θα seems a suitable approach, as they resemble atomic orbitals. However, Gaussian functions are more computationally efficient. Thus, a commonly adopted approach for DFT calculations is using contracted Gaussian functions (CGF) as basis functions, where fixed and computationally efficient Gaussian functions are combined to generate a set of θα to reproduce STOs.
The accuracy and computational cost are usually controlled by changing the size of the basis set. A minimal basis set is the simplest, least accurate and expensive basis set, using only one basis function for each atomic orbital. More accurate and expensive basis set such as double zeta, triple zeta, and quadruple zeta basis are obtained by using two, three, or four basis functions per atomic orbital. As most of the chemically relevant information comes from the valence electrons, a common strategy to improve the computational efficiency with atom centered basis sets is to increase the size of the basis only for the valence orbitals, creating split-valence basis sets. Variational freedom can also be increased by adding polarization and diffuse functions. The first includes functions with higher angular momentum than the valence orbitals of each atom, while the second expands the functions further from the nucleus than regular basis functions.
If the system is placed inside a periodic unit cell, a basis set of plane waves can be used to expand the Kohn–Sham wavefunctions:
where Ω is the volume of the unit cell, defined by the cell vectors a1→, a2→, and a3→, and G→ is a wave vector. Plane-wave basis set are improved systematically, as the number of plane waves can be controlled via a single parameter called kinetic energy cutoff, where a higher kinetic energy cutoff allows higher frequency waves to be added to the calculation and provide a more accurate representation at a higher computational cost.
As the electronic wavefunctions vary rapidly close to the atoms core, an accurate description of such regions using plane-wave basis set requires a large set of functions or equivalently a very large kinetic energy cutoff, increasing the calculation cost. In practical calculations, a method to avoid these difficulties is to model only valence electronic states with the plane-wave basis functions while core states are frozen, forming an effective potential, called a pseudopotential, which combines the core electron states and the nuclei interaction with the electrons in the same term.
A different method to treat this problem is the projector augmented wave or PAW method [11]. Paw performs a linear transformation between all electron and pseudo-wavefunctions, mapping one to another inside a spherical core region centered on the atom and assuring that both wavefunctions will have the same characteristics outside the same core region. The pseudo-wavefunction is smoother inside the core region, it can be well described by plane waves, optimized in situ, and used to recover the behavior of an all electron wavefunction. Thus, PAW offers the possibility of performing calculations with similar computational efficiency as the pseudopotential approach, with a high degree of transferability and provides information about the wavefunctions near the nuclei which can be important for some calculations.
Each basis set and method to treat core electrons has advantages and disadvantages and the choice is typically associated with the system under study and careful convergence tests and comparisons with more expensive calculations or experimental results. Generally, Gaussian basis sets are more appropriate and computationally efficient for small molecules, while plane-wave basis set perform better for periodic and extended systems.
OPTICAL PARAMETRIC DEVICES | Optical Parametric Oscillators (Continuous Wave)
S. Schiller, in Encyclopedia of Modern Optics, 2005
Types of OPOs
OPO types may be classified according to the number of waves that resonate within cavities. Typically, a realizable roundtrip loss coefficient is on the order of Ss = 1% (to which the linear absorption of the nonlinear crystal and mirror transmissivities contribute) and the gain coefficient E ∼ 0.1%/W – 1%/W for Gaussian waves, depending on the pump wavelength, crystal type, etc. This leads to SRO threshold powers on the order of 1 to 10 W. Until the early 1990s, solid-state pump lasers of this power level and single-frequency output were unavailable. This led to the interest in and development of cwOPOs in which more than one of the three waves is resonated (Figure 2). The corresponding external threshold powers are
Here S = T + V is the fractional power attenuation per roundtrip, arising from the mirror transmission T and absorption/scatter loss V. While the mirror transmission maintains power in a given spatial mode, the loss V removes optical power completely and is the fundamental cause for the nonunity efficiency of the frequency conversion process. Typically, each additional resonant enhancement lowers the threshold power by one to two orders.
Examples for threshold values reported in the literature are: 1.9 W for a 920 nm-pumped SRO, 4 mW for a 0.5 μm-pumped DRO, and 140 mW for a 1 μm-pumped common-cavity PR-SRO. DROs can therefore be pumped even by low-power diode lasers. TROs (where pump, signal, and idler are resonated) are not of importance due to their higher complexity.
Optical Fibers
V. Degiorgio, I. Cristiani, in Encyclopedia of Condensed Matter Physics, 2005
Introduction
Immediately after the laser invention, it was realized that laser beams with a carrier frequency of some 1014 Hz have the potential to transport information with a much larger frequency band than either radio or microwaves. Atmospheric transmission is limited to line-of-sight communications and is strongly dependent on visibility conditions. It was, therefore, natural to think about a waveguide having the function of creating a protected path for the light beam, and also of compensating diffraction by optical confinement. Indeed, an optical wave having a finite transversal size broadens during propagation in free space. If one considers the output of a typical laser working at the wavelength λ, this is represented by a Gaussian wave with a beam diameter d ∼1 mm. The diffraction angle θ (also called the divergence of the beam) is of the order of λ/d. Taking λ=1 μm, it is found that at the distance L=1 km the beam diameter becomes 1 m. At present, guided-wave optics is an important technology with applications that are not limited to the transmission of optical signals, but also involve the fabrication of integrated optical and optoelectronic devices, such as lasers and modulators. The basic concept of optical confinement is simple. A dielectric medium of refractive index n1, embedded in a medium of lower refractive index n2, acts as a light trap within which optical rays remain confined by multiple internal reflections at the boundaries. The optical waveguide may have the shape of a slab, strip, or cylinder. The most widely used is the optical fiber, which is made of two concentric cylinders of glass or polymeric material. Silica glass fibers represent, by far, the most important family of fibers, because they can transmit optical pulses over long distances with small attenuation and limited pulse broadening. The fibers made of polymeric material, called plastic optical fibers (POF), although presenting much larger losses than silica fibers, have the advantage of a much cheaper fabrication process, and can be competitive for the short links required in a variety of applications, including communication networks inside a building or a vehicle.
Slater and Gaussian basis functions and computation of molecular integrals
Inga S. Ulusoy, Angela K. Wilson, in Mathematical Physics in Theoretical Chemistry, 2019
2 General representation of molecular orbitals
In most quantum mechanical calculations, it is important to begin a calculation with a good description of the wave function, as this can impact the overall quality of the calculation, which, in turn, impacts the quality of the properties predicted (i.e., bond length, bond angle, dissociation energies, reaction energies). With this in mind, there are numerous choices that can be made for choosing functions that can be used to express molecular orbitals in terms of a set of known functions, and among these are exponential, Gaussian, polynomial, cube, and plane wave functions. However, factors that play an important role in these choices include a desire to ensure that any functional form chosen enables calculations to be feasible, and, preferably, practical, and, that the functions be physically meaningful. In general, molecular orbitals ϕi can be expressed as a linear combination of K basis functions, denoted as χμ, where cμi are coefficients that result in the lowest energy wave function in the Hartree-Fock approach
This is an exact relationship provided that K=∞, resulting in an infinite, or complete, basis set expansion. (A set of basis functions is called a basis set.) This is entirely impractical, as the ab initio Hartree-Fock method scales (in terms of computer time) formally as ≈K4 with the number of basis functions (scaling is also often discussed in terms of molecule size), and this scaling increases quite substantially for other electronic structure methods, which are discussed in a later chapter. (And, when the size of the calculation increases, the computer memory and disk space requirements also increase.)
In considering a basis set which is physically meaningful, ensuring that the set is well defined for any nuclear configuration, and, thus, useful for a theoretical model, a particular set of basis functions can be associated with each nucleus, with dependence upon only the charge of the nucleus. These functions can have the symmetry properties of atomic orbitals, with classifications based on their angular properties (e.g., s, p, d, f, etc. types). This results in a focus upon atomic basis functions, which are the typical form of basis functions used for ab initio calculations. The preceding equation, then, is often referred to as a linear combination of atomic orbitals approach to describe molecular orbitals.
Combining Quantum Mechanics and Molecular Mechanics. Some Recent Progresses in QM/MM Methods
Gustavo Pierdominici-Sottile, ... Juliana Palma, in Advances in Quantum Chemistry, 2010
1 INTRODUCTION
The description of many processes which are relevant to Chemistry and Biology, such as proton and electron transfer, photodissociation, and vibrational energy relaxation, requires the use of quantum mechanics. However, in many cases, these processes take place within large and complex systems, where a full-quantum description cannot be afforded. To cope with such situations, several approximate methods have been developed, each of them having its own strengths and weaknesses. Among them, methods combining quantum and classical mechanics stand out because of their conceptual simplicity and relatively low computational cost.
In this chapter we will concentrate on methods that make explicit use of mixed-quantum/classical (mixed-Q/C) trajectories, designed to analyze processes in real time. Thus, very valuable and highly used approaches such as semiclassical dynamics, Gaussian wave packet propagation (WPP), and path integral will not be reviewed here. Instead, the reader is referred to Refs. [1–4], and the references cited there, for discussions about their characteristics, scope, and applications. Furthermore, this chapter is restricted to electronically adiabatic processes requiring a Q/C treatment because of the presence of nuclear quantum effects, such as the effect of tunneling on a rate constant, or the effect of having discrete energy levels on energy transfer processes. Some of these methods, however, can also be adapted to analyze electronically nonadiabatic processes.
By mixed-Q/C trajectories we mean trajectories in which the evolution of the system is determined through the joint application of the time-dependent Schrödinger equation to some selected degrees of freedom, and the Newton equations to the others. In order to make possible the flow of energy between the two partitions, while maintaining the total energy of the combined system, both quantum and classical equations need to be properly modified. As there is not a single way to do this, different procedures have been proposed which can be distinguished by the strategy used to connect the quantum and the classical subsystems.
One of the most useful features of the methods based on mixed-Q/C trajectories is that they allow observing processes in real time. Thus, a clear picture of their dynamics can be readily appreciated. This is particularly helpful in studies of vibrational energy relaxation and intramolecular energy redistribution. Besides, statistical results such as reaction probabilities or canonical rate constants can also be obtained through averaging over a large number of trajectories.
In the following section we briefly present the basic equations of the most used mixed-Q/C propagation schemes. This is mainly done to introduce the terminology and equations employed in the discussions given in the rest of the chapter. For more detailed descriptions, as well as comparisons between the different schemes, the reader is referred to Refs. [5–7]. Then, we discuss the application of mixed-Q/C trajectories to the study of three different kinds of processes, selected among the large variety that can be analyzed with this methodology. These processes are (1) hydrogen transfer reactions in gas phase; (2) hydride/proton transfer reactions in condensed phase; and (3) vibrational energy relaxation of small solutes in van der Waals clusters and condensed phase. We believe that, by discussing the applications of mixed-Q/C trajectories to these selected cases, the capabilities and limitations of the approach will be clearly exemplified. We conclude this chapter with a discussion on prospective implementations of this methodology.
Recommended publications
Computational Materials ScienceJournal
Applied Ocean ResearchJournal
Optics & Laser TechnologyJournal
Journal of Sound and VibrationJournal
https://www.sciencedirect.com/topics/engineering/gaussian-wave
Gaussian Wave - an overview | ScienceDirect Topics
In most quantum mechanical calculations, it is important to begin a calculation with a good description of the wave function, as this can impact the overall quality of the calculation, which, in turn, impacts the quality of the properties predicted (i.e.,
www.sciencedirect.com
'학문 > 물리' 카테고리의 다른 글
| 총알의 충격 (저격총 발포 총알의 파괴력) (0) | 2022.11.29 |
|---|---|
| [양자역학] 2.2-(1) Example: 파동 패킷 Wave Packet (0) | 2022.04.08 |
| 파동 (0) | 2022.04.08 |
| Tensor에 관한 기초 설명 (0) | 2020.10.18 |
| 테넷에 대한 어떤 사람의 다양한 이야기 (0) | 2020.09.24 |
