Quantum level

Classical molecular simulation methods such as MC and MD represent atomistic/molecular-level modeling, which discards the electronic degrees of freedom while utilizing parameters transferred from quantum level simulation as force field information. A molecule in the simulation is composed of beads representing atoms, where the interactions are described by classical potential functions. Each bead has a dispersive pair-wise interaction as described by the Lennard-Jones (LJ) potential, ULj(Ly)  

rib denotes the interbead distance (i.e., the bond length between two adjacent beads), k is the spring constant that quantifies the rigidity of the bond, and R0 is the maximum extensibility of the spring. The form of torsional potential parameters, describing four bonded atoms, is 

The molecular motion in MD simulation is deterministic by solving a Hamiltonian system (Allen and Tildesley, 1996). For the precise description of the polymeric systems, Langevin dynamics (Grest, 1996) were employed, where the force acting on the z th bead in the ath molecule can be calculated by the following equation  

m and rai are the mass and position vector of beads, respectively. is the friction tensor, which is assumed to be isotropic for simplicity in our simulation, that is, = Fl, where I is the unit dyad and r = 0.5t 1 (t = cr(m/ )° 5j (Grest, 1996). Further, f aj is the Brownian random force, which obeys the Gaussian white noise, and is generated according to the fluctuation—dissipation theorem  

- is the bead momentum vector and u(rm. f) = iyrV is the linear streaming velocity profile, where y = dux/dy is the shear strain rate. Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)). 

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However, the reader may be wondering, what is the connection of all of these classical notions—stable nonnal modes, regular motion on an invariant toms—to the quantum spectmm of a molecule observed in a spectroscopic experiment Recall that in the hannonic nonnal modes approximation, the quantum levels are defined by the set of quantum numbers (Up. . Uyy) giving the number of quanta in each of the nonnal modes. 

The other kind of quantum level we considered above is one with quanta in more than one mode, i.e. n. . ... 

Real molecules in general have many quantum levels, and the TDSE can exliibit complicated behaviour even in the absence of a field. To simplify matters, it is worthwhile discussing some properties of the solutions of the TDSE in the absence of a field and then reintroducing the field. First let us consider... 

Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

Figure A3.13.14. Illustration of the quantum evolution (pomts) and Pauli master equation evolution (lines) in quantum level structures with two levels (and 59 states each, left-hand side) and tln-ee levels (and 39 states each, right-hand side) corresponding to a model of the energy shell IVR (liorizontal transition in figure...

Figure A3.13.15. Master equation model for IVR in highly excited The left-hand side shows the quantum levels of the reactive CC oscillator. The right-hand side shows the levels with a high density of states from the remaining 17 vibrational (and torsional) degrees of freedom (from [38]).

Considerable spectroscopic data are required for the detemiination of the relative populations in die various internal quantum levels of the product from the relative intensities of various lines, or bands, in a spectrum. 

Note. The maximum number of electrons that any quantum level can accommodate is seen to be given by the formula 2n where n is the number of the quantum level, for example = 3 the maximum number of electrons is therefore 18. 

For the next elements, sodium to argon, the = 3 quantum level fills up in the same way as then = 2quantum level. This is shown in Table 1.3. 

Reference to the modern periodic table (p. (/)) shows that we have now completed the first three periods—the so-called short periods. But we should note that the n = 3 quantum level can still accommodate 10 more electrons. 

Note. The electronic configuratioa of any element can easily be obtained from the periodic table by adding up the numbers of electrons in the various quantum levels. We can express these in several ways, for example electronic configuration of nickel can be written as ls 2s 2p 3s 3d 4s. or more briefly ( neon core ) 3d 4s, or even more simply as 2. 8. 14. 2... 

Chemical properties and spectroscopic data support the view that in the elements rubidium to xenon, atomic numbers 37-54, the 5s, 4d 5p levels fill up. This is best seen by reference to the modern periodic table p. (i). Note that at the end of the fifth period the n = 4 quantum level contains 18 electrons but still has a vacant set of 4/ orbitals. 

Except for the n = 1 quantum level the maximum number of electrons in the outermost quantum level ofany period isalwayseight. At this point the element concerned is one of the noble gases (Chapter 12). 

The table contains vertical groups of elements each member of a group having the same number of electrons in the outermost quantum level. For example, the element immediately before each noble gas, with seven electrons in the outermost quantum level, is always a halogen. The element immediately following a noble gas, with one electron in a new quantum level, is an alkali metal (lithium, sodium, potassium, rubidium, caesium, francium). 

The periodic table also contains horizontal periods of elements, each period beginning with an element with an outermost electron in a previously empty quantum level and ending with a noble gas. Periods 1, 2 and 3 are called short periods, the remaining are long periods Periods 4 and 5 containing a series of transition elements whilst 6 and 7 contain both a transition and a rare earth senes,... 

The number of electrons in the outermost quantum level of an atom increases as we cross a period of typical elements. Figure 2.2 shows plots of the first ionisation energy for Periods 2 and 3,... 

There are many compounds which do not conduct electricity when solid or fused indicating that the bonding is neither metallic nor ionic. Lewis, in 1916. suggested that in such cases bonding resulted from a sharing of electrons. In the formation of methane CH4 for example, carbon, electronic configuration l.s 2.s 2p. uses the tour electrons in the second quantum level to form four equivalent... 

These apparent anomalies are readily explained. Elements in Group V. for example, have five electrons in their outer quantum level, but with the one exception of nitrogen, they all have unfilled (I orbitals. Thus, with the exception of nitrogen. Group V elements are able to use all their five outer electrons to form five covalent bonds. Similarly elements in Group VI, with the exception of oxygen, are able to form six covalent bonds for example in SF. The outer quantum level, however, is still incomplete, a situation found for all covalent compounds formed by elements after Period 2. and all have the ability to accept electron pairs from other molecules although the stability of the compounds formed may be low. This... 

In Group III, boron, having no available d orbitals, is unable to fill its outer quantum level above eight and hence has a maximum covalency of 4. Other Group 111 elements, however, are able to form more than four covalent bonds, the number depending partly on the nature of the attached atoms or groups. 

The ability to act as a lone pair acceptor is not confined to Group III, and can occur wherever a quantum level is incomplete. This ability to accept electrons explains why covalent chlorides, with the exception of carbon tetrachloride, are readily hydrolysed, the apparently anomalous behaviour of carbon tetrachloride being readily explained by the fact that the carbon has a completed quantum level and is unable to form an intermediate complex with water. 

Boron achieves a covalency of three by sharing its three outer electrons, for example BFj (p. 153). By accepting an electron pair from a donor molecule or ion, boron can achieve a noble gas configuration whilst increasing its covalency to four, for example H3N->BCl3. K BF4. This is the maximum for boron and the second quantum level is now complete these 4-coordinate species are tetrahedral (p. 38). 

In this group the outer quantum level has a full s level and two electrons in the corresponding p level. As the size of the atom increases the ionisation energy changes (see Table 8.1) and these changes are reflected in the gradual change from a typical non-metallic element, carbon, to the weakly metallic element, lead. Hence the oxides of carbon and silicon are acidic whilst those of tin and lead are amphoteric. 

The concept of oxidation states is best applied only to germanium, tin and lead, for the chemistry of carbon and silicon is almost wholly defined in terms of covalency with the carbon and silicon atoms sharing all their four outer quantum level electrons. These are often tetrahedrally arranged around the central atom. There are compounds of carbon in which the valency appears to be less than... 

Carbon, however, is unable to form similar complexes since the energy required to promote electrons to the next higher energy level, the 3s, is too great (or since carbon has no available d orbitals in its outer quantum level). 

When carbon forms four covalent bonds with halogen atoms the second quantum level on the carbon is completely filled with electrons. Most of the reactions of the Group IV tetrahalides require initial donation by a Lewis base (p. 91) (e.g. water, ammonia) which attaches initially to the tetrahalide by donation of its electron pair. Hence, although the calculated free energy of a reaction may indicate that the reaction is energetically favourable, the reaction may still not proceed. Thus we find that the tetrahalides of carbon... 

The elements in this group have six electrons in their outer quantum level, and can thus achieve a noble gas configuration by acquiring two electrons. 

The molecule of sulphur dioxide has a bent structure. Both S—O distances are equal and short and since sulphur can expand its outer quantum level beyond eight, double bonds between the atoms are likely i.e. 

Numerous ionic compounds with halogens are known but a noble gas configuration can also be achieved by the formation of a covalent bond, for example in halogen molecules, X2, and hydrogen halides, HX. When the fluorine atom acquires one additional electron the second quantum level is completed, and further gain of electrons is not energetically possible under normal circumstances, i.e... 

The large ionisation energies, as expected for atoms with com plete quantum levels. 

Reference has been made already to the existence of a set of inner transition elements, following lanthanum, in which the quantum level being filled is neither the outer quantum level nor the penultimate level, but the next inner. These elements, together with yttrium (a transition metal), were called the rare earths , since they occurred in uncommon mixtures of what were believed to be earths or oxides. With the recognition of their special structure, the elements from lanthanum to lutetium were re-named the lanthanons or lanthanides. They resemble one another very closely, so much so that their separation presented a major problem, since all their compounds are very much alike. They exhibit oxidation state -i-3 and show in this state predominantly ionic characteristics—the ions. 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

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