Electric dipole, definition

At the molecular level, electric quadrupoles can lead to useful structural information. Thus, whilst the absence of a permanent electric dipole in CO2 simply means that the molecule is linear, the fact that the electric quadrupole moment is negative shows that our simple chemical intuition of 0 C" 0 is correct. The definition of quadrupole moment is only independent of the coordinate origin when the charges sum to zero and when the electric dipole moment is zero. 

Conjugated polymers are centrosymmetric systems where excited states have definite parity of even (A,) or odd (B ) and electric dipole transitions are allowed only between states of opposite parity. The ground state of conjugated polymers is an even parity singlet state, written as the 1A... PM spectroscopy is a linear technique probing dipole allowed one-photon transitions. Non linear spectroscopies complement these measurements as they can couple to dipole-forbidden trail-... 

Any molecule has an infinity of excited orbitals in the continuum above the first ionization energy. The electric dipole polarizability is connected partly with a few of these continuum orbitals and partly with the valence orbitals (7). If the simultaneous formation of empty orbitals of X, but with the continuum, it is reasonable to think of M being polarized by X. The population of the continuum orbitals of X is expected to be the more... 

In resolving the apparent paradox of how an antenna with no charge (in free space) can have an electric-dipole moment, one can go back to definitions. In Ref. 14 the fields from a current distribution are evaluated by expanding... 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

We can use parity to aid in determining selection rules. Recall (Section 1.8) that the integral vanishes if the integrand is an odd function of the Cartesian coordinates. The operator d [Equation (1.286)] is an odd function. If the wave functions are of definite parity, as is usually true, then if states m and have the same parity, the integrand in mn will be odd. Hence electric-dipole transitions are forbidden between states of the same parity we have the selection rule parity changes. (This is the Laporte rule.)... 

Another example is the particle in a box. With the origin at the center of the box, the potential energy is an even function, and the wave functions are of definite parity, determined by whether the quantum number is odd or even. Hence for electric-dipole transitions, the quantum number must go from even to odd, or vice versa, as concluded previously. 

The definitive resonance Raman spectra obtained by Clark and Franks for [(C4H,)4N] 2Re2Cl8 and [(C4H,)4N]2Re2Br8 were particularly informative (49). The observation of resonance enhancement in the Raman spectrum recorded with excitation frequencies in the range of the lowest electronic absorption near 14,000 cm 1 not only provided useful vibrational data, but also confirmed that this electronic absorption was electric dipole allowed in accord with the S- 5 assignment. Normal Raman spectra were obtained with excitation energies differing substantially from the absorption maximum near... 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

P is the macroscopic polarization. It consists of a lattice polarization b21 w originating from the electric dipole moment arising from the mutual displacement of the two sublattices, and of a second term b22 P originating from the pure electron polarization. According to definition, P and E are connected by... 

In quantum mechanics the definition of molecular polarizabilities is given through time-dependent perturbation theory in the electric dipole approximation. These expressions are usually given in terms of sums of transition matrix elements over energy denominators involving the full electronic structure of the molecule [42]. 

Table 8. Increase in electric dipole moments upon formation of hydrogen-bonded complexes (for the definition of Ap see Eq. (7))...

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

We shall calculate the matrix elements of the effective Hamiltonian within the basis of these six primitive states in due course. These states do not, however, have definite parities. Since parity is conserved (except in the presence of an applied electric field), we construct a basis set of six functions, three of each parity type, so that for a given J level we are left with the diagonalisation of 3 x 3 matrices, rather than 6x6. More importantly, we are aiming to understand the electric dipole radio frequency and microwave spectra, and know that transitions must occur between states of opposite parity. 

These two functions do not have definite parities but the symmetric and antisymmetric combinations of them do we use these combinations to calculate both the Stark effect and the electric dipole transition probabilities. 

On inserting these expressions into equation (35), we obtain the general definition of the moment of the electric dipole for an arbitrary microsystem ... 

Some molecular tensors (electric dipole polarizability, electric and magnetoelectric shielding) are origin independent, as can be immediately found by inspection of definitions (87)-(112). Other tensors depend on the origin assumed for the multipole expansion. For instance, in a change of origin... 

We can give a simple, but important physical interpretation of the expressions for sensitivities, (9.55), based on the reciprocity principle. Note that, according to definition (see Chapter 8), the Green s tensors (r r") and Gh (r r"), are the electric and magnetic fields at the receiver point, r, due to a unit electric dipole source at the point r" of the conductivity perturbation. Let us introduce a Cartesian system of coordinates x,y,z, and rewrite these tensors in matrix form ... 

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

Now consider the set of Stokes operators that can be obtained by canonical quantization of (132). On the other hand, the Stokes operators should by definition represent the complete set of independent Hermitian bilinear forms in the photon operators of creation and annihilation. It is clear that such a set is represented by the generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of electric dipole photons. The nine generators have the form [46]... 

Finally starting from the usual definitions of atomic electric dipole and quadrupole and magnetic dipole moments fAkel, qfcel, and jnkm... 

The difference between the two definitions rests with the fact that a pyroelectric crystal must possess an overall (observable) permanent electric dipole. Thus a pyroelectric crystal is built from unit cells, each of which must contain an overall electric dipole, (Figure 4.11b). The pyroelectric effect will only be observed, however, if all of these dipoles are aligned throughout the crystal, (Figure 4.11c). [Note that a ferroelectric crystal is defined in a similar way. The difference between a pyroelectric crystal and a ferroelectric crystal lies in the fact that the direction of each overall electric dipole in a ferroelectric crystal can be altered by an external electric field.]... 

A ferroelectric crystal is one that has an electric dipole moment even in the absence of an external electric field. This arises because the centre of positive charge in the crystal does not coincide with the centre of negative charge. The phenomenon was discovered in 1920 by J. Valasek in Rochelle salt, which is the H-bonded hydrated d-taitrate NaKC4H406.4H20. In such compounds the dielectric constant can rise to enormous values of 10 or more due to presence of a stable permanent electric polarization. Before considering the effect further, it will be helpful to recall various definitions and SI units ... 

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