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According to the Pauli exclusion principle, the total wave function describing the entire system of N electrons must be fully antisymmetric in the (spatial and spin) coordinates of the electrons, and will therefore be a Slater determinant constructed from the individual spin orbitals [7.34]. The corresponding total e... | {
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Since the maximum kinetic energy of an electron in a Fermi electron gas at 0 K is the Fermi energy $E_F$ , we write for the total energy of the most energetic electrons of the system the classical equation
$$E_{\text{max}} = E_{\text{F}} + V(r)$$
[7.52]
It is clear that $E_{\rm max}$ must be independent of r, be... | {
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The Thomas-Fermi equation [7.66] is a 'universal' equation, which does not depend on Z, nor on physical constants such as $\hbar$ , m or e which have been 'scaled out' by performing the change of variables [7.63]. We also note that it is a second-order, non-linear equation. Since the boundary condition at the origin... | {
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When $r \to \infty$ , we see from [7.71] and [7.73] that $rV(r) \to 0$ , so that the Thomas-Fermi potential [7.73] falls off more rapidly than 1/r for larger r. This behaviour is at variance with the result [7.9c] which we obtained in our discussion of the central field approximation. The reason is that the potential... | {
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We shall now study a more elaborate approximation for complex atoms (ions), known as the Hartree-Fock or self-consistent field method. The starting point of this approach, formulated by Hartree in 1928, is the independent particle model, discussed in Section 7.1, according to which each electron moves in an effective p... | {
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"Header 3": "7.4 THE HARTREE-FOCK METHOD AND THE SELF-CONSISTENT FIELD",
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We first have
$$\langle \Phi | \hat{H}_{1} | \Phi \rangle = N! \langle \Phi_{H} | \mathcal{A} \hat{H}_{1} \mathcal{A} | \Phi_{H} \rangle$$
$$= N! \langle \Phi_{H} | \hat{H}_{1} \mathcal{A}^{2} | \Phi_{H} \rangle$$
$$= N! \langle \Phi_{H} | \hat{H}_{1} \mathcal{A} | \Phi_{H} \rangle$$
[7.95]
where we have used [... | {
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"Header 3": "7.4 THE HARTREE-FOCK METHOD AND THE SELF-CONSISTENT FIELD",
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We note that both $\mathcal{J}_{\lambda\mu}$ and $K_{\lambda\mu}$ are real; they are also symmetric in $\lambda$ and $\mu$ ,
$$\mathcal{J}_{\lambda\mu} = \mathcal{J}_{\mu\lambda} \qquad K_{\lambda\mu} = K_{\mu\lambda} \tag{7.105}$$
In terms of $\mathcal{J}_{\lambda\mu}$ and $K_{\lambda\mu}$ , [7.102] read... | {
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"Header 3": "7.4 THE HARTREE-FOCK METHOD AND THE SELF-CONSISTENT FIELD",
"token_count": 2019,
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We also define the exchange (non-local) operator $V_{\mu}^{\rm ex}(q_i)$ such that
$$V_{\mu}^{\text{ex}}(q_i)f(q_i) = \left[ \int u_{\mu}^{\star}(q_j) \frac{1}{r_{ij}} f(q_j) \, \mathrm{d}q_j \right] u_{\mu}(q_i)$$
[7.117]
where $f(q_i)$ is an arbitrary function. In particular, when acting on a spin-orbital $u... | {
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"Header 3": "7.4 THE HARTREE-FOCK METHOD AND THE SELF-CONSISTENT FIELD",
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A striking feature of the Hartree-Fock equations [7.129] is that they look similar to individual Schrödinger eigenvalue equations for each of the spin orbitals $u_{\lambda}$ . They are not genuine eigenvalue equations, however, since the Hartree-Fock potential $\mathcal{V}$ depends on the spin-orbitals themselves th... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"token_count": 1986,
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We may also rewrite [7.140] as
$$E[\Phi] = \sum_{\lambda} E\lambda - \langle \Phi | \hat{H}_2 | \Phi \rangle$$
[7.141]
and we see that the total energy is *not* the sum of the individual energies. This is because in summing the individual electron energies, each kinetic energy and each interaction energy with the n... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"token_count": 1994,
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Using the addition theorem of the spherical harmonics (see equation [A4.23] of Appendix 4) we have
$$\sum_{m'=-l'}^{+l'} |Y_{l'm'}(\theta_j, \phi_j)|^2 = \frac{2l'+1}{4\pi}$$
[7.150]
so that
$$\mathcal{V}_{n'l'}^{d} = \frac{2(2l'+1)}{4\pi} \int |P_{n'l'}(r_j)|^2 \frac{1}{r_{ij}} dr_j d\Omega_j$$
[7.151]
The int... | {
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"token_count": 1943,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In this case the Slater determinant [7.87] reads
$$\Phi_{1,1}, q_{2}, q_{3}, q_{4}) = \frac{1}{\sqrt{4!}} \begin{vmatrix} u_{1s} \uparrow(q_{1}) & u_{1s} \downarrow(q_{1}) & u_{2s} \uparrow(q_{1}) & u_{2s} \downarrow(q_{1}) \\ u_{1s} \uparrow(q_{2}) & u_{1s} \downarrow(q_{2}) & u_{2s} \uparrow(q_{2}) & u_{2s} \downar... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"token_count": 2040,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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However, in contrast to the hydrogenic radial functions [3.53], the Slater orbitals do not possess radial nodes. In terms of Slater orbitals $\chi_i(\mathbf{r})$ , a Hartree-Fock spatial orbital $u(\mathbf{r})$ is then given by
$$u(\mathbf{r}) = \sum_{i=1}^{N} c_i \chi_i(\mathbf{r})$$
[7.168]
where the quantitie... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"token_count": 2002,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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It is also worth noting that in contrast with the Hartree–Fock equations, where all the electrons move in the same Hartree–Fock potential for a given state of the system, the effective potential in the Hartree equations depends on the particular orbital $u_{\lambda}$ considered. It follows that in general the Hartree... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"token_count": 1198,
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In the central field approximation, the non-relativistic Hamiltonian [7.2] of the N-electron atom (ion) is replaced by the Hamiltonian
$$H_{c} = \sum_{i=1}^{N} h_{i}, \qquad [7.178a]$$
where $h_i$ is the individual Hamiltonian of electron i in the central field $V(r_i)$ (see
[7.11]). For example, if we choose... | {
"Header 1": "Physical interpretation of the Hartree-Fock equations. Self-consistent field. Koopman's theorem",
"Header 2": "7.5 CORRECTIONS TO THE CENTRAL FIELD APPROXIMATION. L-S COUPLING AND j-j COUPLING",
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In order to determine all the possible terms corresponding to a given configuration (that is, the possible values of L and S), the rules for the addition of angular momenta must be used. However, in combining the individual electron orbital angular momenta $L_i$ to obtain L and the individual electron spin angular mo... | {
"Header 1": "Determination of the possible terms of a multielectron configuration in L-S coupling",
"token_count": 1943,
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The determination of the possible terms for equivalent 'optically active' electrons is more difficult. Indeed, in this case certain values of L and S are ruled out because of the Pauli exclusion principle.
The simplest case is that of two equivalent s electrons, corresponding to the configuration $ns^2$ . As we have... | {
"Header 1": "Determination of the possible terms of a multielectron configuration in L-S coupling",
"Header 3": "Electrons belonging to the same subshell (equivalent electrons)",
"token_count": 1945,
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possible terms for electron configurations $(nl)^k$ , with l=0,1,2
| Configurat | ion | | ... | {
"Header 1": "Determination of the possible terms of a multielectron configuration in L-S coupling",
"Header 3": "Electrons belonging to the same subshell (equivalent electrons)",
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Having obtained the energy levels of the Hamiltonian [7.184], we now proceed to the second step of the perturbation calculation, which consists in taking into account the spin-orbit term $H_2$ given by [7.179]. We shall first examine how the additional perturbation $H_2$ further removes degeneracies. The total Hami... | {
"Header 1": "Fine structure of terms in L-S coupling. Multiplet splitting and the Lande interval rule",
"token_count": 1850,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Since $J^2 = L^2 + 2L \cdot S + S^2$ , we then have (with $A = \overline{A}\hbar^2$ )
$$\langle \gamma L S \mathcal{J} M_{\mathcal{J}} | H_2 | \gamma L S \mathcal{J} M_{\mathcal{J}} \rangle = \frac{1}{2} \overline{A} \langle \gamma L S \mathcal{J} M_{\mathcal{J}} | \mathbf{J}^2 - \mathbf{L}^2 - \mathbf{S}^2 | \gamm... | {
"Header 1": "Fine structure of terms in L-S coupling. Multiplet splitting and the Lande interval rule",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In the case of j-j coupling the notation for the spectral terms must specify the quantum numbers $(n_i l_i j_i)$ of each electron and the total angular momentum quantum number $\mathcal{J}$ . The values of the individual $j_i$ 's are usually written between parentheses, and $\mathcal{J}$ as a subscript. For exa... | {
"Header 1": "Fine structure of terms in L-S coupling. Multiplet splitting and the Lande interval rule",
"token_count": 1609,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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(b) Obtain the Thomas-Fermi equation [7.66] by minimising the expression [1], subject to the normalisation condition
$$4\pi \int_0^\infty \rho(r)r^2 dr = Z$$
- 7.4 Using the Thomas-Fermi model, obtain an estimate of the following quantities:
- (a) average distance of an electron from the nucleus;
- (b) average ki... | {
"Header 1": "Fine structure of terms in L-S coupling. Multiplet splitting and the Lande interval rule",
"token_count": 759,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In the previous two chapters, we discussed the energy levels of atoms with more than one electron and now we describe some features of the spectra that result from transitions between these levels, and also how the levels are perturbed by external static electric and magnetic fields. For the most part, this is a straig... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"token_count": 1809,
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Similarly the spherical components of the polarisation vector $\hat{\epsilon}$ are $\epsilon_q$ , where (see [4.81])
$$\varepsilon_1 = -\frac{1}{\sqrt{2}} (\hat{\varepsilon}_x + i\hat{\varepsilon}_y), \ \varepsilon_0 = \hat{\varepsilon}_z, \ \varepsilon_{-1} = \frac{1}{\sqrt{2}} (\hat{\varepsilon}_x - i\hat{\var... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Thus the (2s) level in lithium lies well below the (2p) level, and for sodium the (3s) level is below the (3p) level, which in turn is below the (3d) level (see Fig. 8.1). As the excitation increases and the levels become more hydrogenic in character, there is near degeneracy in l, for a given n. The valence electron i... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"token_count": 2030,
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levels of such a sequence can be written as
$$E_{nl} = -\frac{1}{2} \frac{\tilde{Z}^2}{[n - \alpha(l)]^2} \text{a.u.}$$
[8.21]
where $\tilde{Z} = Z - N + 1$ , Z being the nuclear charge and N the number of electrons. The quantity $\tilde{Z}$ is therefore the net charge of the nucleus and the core electrons, an... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"token_count": 1915,
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The spin magnetic dipole moment of each electron is $-2\mu_B \mathbf{S}_i/\hbar$ (i=1,2) and the interaction energy between them, when separated by a distance $r \neq 0$ , is
$$V_{S}(r) = \frac{\mu_{0}}{4\pi} \frac{4\mu_{B}^{2}}{\hbar^{2}} \left[ \frac{\mathbf{S}_{1} \cdot \mathbf{S}_{2}}{r^{3}} - 3 \frac{(\mathbf... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"token_count": 1667,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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The spectra of atoms with one and two valence electrons consist, as we have seen, of simple series and the same is largely true of the trivalent elements such as B and Al. Such simple series can be readily identified and analysed, but in the complex spectra of more complicated systems, series are much more difficult, a... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"Header 3": "8.4 ATOMS WITH SEVERAL OPTICALLY ACTIVE ELECTRONS. MULTIPLET STRUCTURE",
"token_count": 2021,
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The series limit $(n \rightarrow \infty)$ , is at a higher energy than the series limit for the regular series $(1s)^2(2s)p \rightarrow (1s)^2(2s)^2$ , so the terms are said to be displaced. Transitions between the regular and displaced terms are possible. For example in Be, there is a line at $14320 \text{ cm}^{-1}... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"Header 3": "8.4 ATOMS WITH SEVERAL OPTICALLY ACTIVE ELECTRONS. MULTIPLET STRUCTURE",
"token_count": 1968,
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In the absence of external fields, there is no preferred direction in space and the energy of an isolated system, such as an atom, cannot depend on which direction we choose as the Z axis. As a consequence, the energy of an atom does not depend on $M_{\mathcal{I}}\hbar$ , the eigenvalues of $\mathcal{I}_z$ , and the ... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"Header 3": "8.5 INTERACTION WITH MAGNETIC FIELDS. THE ZEEMAN EFFECT",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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This is the Back-Goudsmit effect and the theory can be developed in a similar way to that of the Paschen-Back effect. The interaction $\bar{C}(\mathbf{I} \cdot \mathbf{J})$ is first omitted. In this case the atom is in an eigenstate of $\mathbf{J}^2$ , $\mathcal{J}_z$ and of $\mathbf{I}^2$ , $I_z$ which is $(2... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"Header 3": "8.5 INTERACTION WITH MAGNETIC FIELDS. THE ZEEMAN EFFECT",
"token_count": 2026,
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A perturbed ${}^2P_{\mathcal{T}}$ wave function, for example, now contains a small admixture of the ${}^2D_{\mathcal{T}}$ function. As a consequence, lines are observed in the presence of an electric field that would normally be forbidden, for example the series $n^2S \to n'^2D$ and $n^2S \to n'^2S$ .
#### 8.7... | {
"Header 1": "The interaction of many-electron atoms with electromagnetic fields",
"Header 3": "8.5 INTERACTION WITH MAGNETIC FIELDS. THE ZEEMAN EFFECT",
"token_count": 2044,
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Bound systems of electrons and more than one nucleus are known as molecules. In this chapter, we shall discuss the structure of molecules, concentrating on the simplest diatomic systems which contain just two nuclei. We shall start by discussing the general nature of molecular structure, showing how the rotational, vib... | {
"Header 1": "9 Molecular structure",
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To see how the ideas of the previous paragraph translate into the language of quantum mechanics, we shall consider a diatomic molecule composed of nuclei A and B, of masses $M_A$ and $M_B$ , together with a number N of electrons. The internuclear coordinate will be denoted by **R** and the position vectors of the el... | {
"Header 1": "9 Molecular structure",
"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
"token_count": 1586,
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That is,
$$\sum_{q} \int d\mathbf{r}_{1} d\mathbf{r}_{2} \cdot \cdot \cdot d\mathbf{r}_{N} \, \Phi_{s}^{\star} [T_{N} + T_{e} + V - E] F_{q}(\mathbf{R}) \Phi_{q} = 0,$$
$$s = 1, 2, \dots \qquad [9.12]$$
Using the equation [9.9] satisfied by the functions $\Phi_q$ and the orthonormality property [9.10], the coup... | {
"Header 1": "9 Molecular structure",
"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
"token_count": 1941,
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That is
$$E_r = \frac{\hbar^2}{2\mu R_0^2} \mathcal{J}(\mathcal{J} + 1)$$
$$= \frac{\hbar^2}{2I_0} \mathcal{J}(\mathcal{J} + 1)$$
$$= B\mathcal{J}(\mathcal{J} + 1) \qquad \mathcal{J} = 0, 1, 2, \dots$$
[9.21]
where $I_0 = \mu R_0^2$ is the moment of inertia for the reduced mass $\mu$ and the equilibrium dis... | {
"Header 1": "9 Molecular structure",
"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
"token_count": 1827,
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Adopting the Morse potential [9.23] for $E_s(R) - E_s(\infty)$ , this equation reads
$$\left[-\frac{\hbar^2}{2\mu}\frac{\mathrm{d}^2}{\mathrm{d}R^2}+V_{\mathrm{eff}}(R)-\bar{E}_{s,v,\mathcal{J}}\right]\mathcal{F}_{v,\mathcal{J}}^s(R)=0$$
[9.32]
where $\bar{E}_{s,v,\mathcal{J}} = E_{s,v,\mathcal{J}} - E_s(\infty)$... | {
"Header 1": "9 Molecular structure",
"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
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#### 9.4 ELECTRONIC STRUCTURE OF DIATOMIC MOLECULES
In this section we shall discuss the electronic wave functions of the simplest diatomic molecules. Our discussion will be based on the electronic wave equation [9.9], where we recall that all but the Coulomb interactions are neglected in the electronic Hamiltonian... | {
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"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
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We now turn to the question of the behaviour of the electronic wave function of a homonuclear diatomic molecule if the two identical nuclei are interchanged, so that $\mathbf{R} \rightarrow -\mathbf{R}$ and
$$\Phi_{s}(\mathbf{R}; \mathbf{r}_{1}, \mathbf{r}_{2}, \dots \mathbf{r}_{N}) \to \Phi_{s}(-\mathbf{R}; \mathb... | {
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"Header 3": "9.2 THE BORN-OPPENHEIMER SEPARATION FOR DIATOMIC MOLECULES",
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The electronic terms or potential curves $E_s(R)$ of a diatomic molecule depend only on the internuclear distance R, and it is important to investigate the behaviour of these potential curves as R varies. We shall analyse below a few low-lying potential curves of simple molecular systems such as $H_2^+$ and $H_2$ ... | {
"Header 1": "9 Molecular structure",
"Header 2": "Intersection of potential curves and the von Neumann-Wigner non-crossing rule",
"token_count": 1952,
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Although these functions are only expected to be accurate in the asymptotic region of large R, we can use them as trial functions in the variational expression [2.361], namely
$$E_{g,u}(R) = \frac{\int \Phi_{g,u}^{\star} H \Phi_{g,u} d\mathbf{r}}{\int |\Phi_{g,u}|^2 d\mathbf{r}}$$
[9.48]
Let us first work out the d... | {
"Header 1": "9 Molecular structure",
"Header 2": "Intersection of potential curves and the von Neumann-Wigner non-crossing rule",
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with $Z^*$ a function of R, and by determining $Z^*$ at each R with the help of the Rayleigh-Ritz variational method (see Chapter 2). We also note that at large R a dipole moment is induced in a hydrogen atom by the electrostatic field of a proton. This interaction gives rise to a potential proportional to $R^{-... | {
"Header 1": "9 Molecular structure",
"Header 2": "Intersection of potential curves and the von Neumann-Wigner non-crossing rule",
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Four combinations can be formed, namely
$$\Phi_{A}(1, 2) = \Phi_{g}(1)\Phi_{g}(2)\chi_{0,0}(1, 2)$$
[9.67a]
$$\Phi_{\rm B}(1,2) = \Phi_{\rm u}(1)\Phi_{\rm u}(2)\chi_{0.0}(1,2)$$
[9.67b]
$$\Phi_{\rm C}(1, 2) = \frac{1}{\sqrt{2}} \left[ \Phi_{\rm g}(1) \Phi_{\rm u}(2) + \Phi_{\rm g}(2) \Phi_{\rm u}(1) \right] \chi_... | {
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That is,
$$\Phi_{\rm T} = \Phi_{\rm A} + \lambda \Phi_{\rm B} \tag{9.79}$$
The parameter $\lambda$ can be determined by the variational method. We first obtain the energy as a function of $\lambda$ (for a fixed value of R):
$$E(\lambda) = \frac{\int d\mathbf{r}_1 d\mathbf{r}_2 \, \Phi_T^{\star} H \Phi_T}{\int... | {
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9.14 Relationships between the ground state energies of $H_2$ , $H_2^+$ and H. The chemical dissociation energies and spectroscopic dissociation energies denoted by $D_0$ and $D_e$ , respectively, differ by the zero-point vibrational energy $\frac{1}{2}\hbar\omega_0$ . The ionisa... | {
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Three of the p electrons in each atom can form the bonding orbitals $(\sigma_g 2p)^2(\pi 2p_x)^2(\pi 2p_y)^2$ . The remaining pair of electrons must be associated with antibonding orbitals $(\pi^*2p)$ . It turns out that one electron goes into the $(\pi^*2p_x)$ and one into the $(\pi^*2p_y)$ orbital.
#### Pairi... | {
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The ground state of the molecule, which has two electrons in each of the $1\sigma$ and $2\sigma$ orbitals, becomes $Li^+ + H^-$ in the separated atom limit, with two electrons in the 1s orbital of $Li^+$ and two in the 1s orbital of $H^-$ . At the equilibrium distance, $R_0 = 1.6$ Å, excess negative charge... | {
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The rotational motion may be treated approximately by supposing that the nuclei are fixed at their equilibrium position, so that the molecule forms a rigid body. If the molecules possesses an n-fold symmetry axis, with $n \ge 3$ , then two of the three principal moments of inertia [7] of the rigid body are equal, and ... | {
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Four combinations can be constructed:
$$\begin{split} & \Phi_1 = v_{2s} + v_{2p_x} + v_{2p_y} + v_{2p_z} \\ & \Phi_2 = v_{2s} + v_{2p_x} - v_{2p_y} - v_{2p_z} \\ & \Phi_3 = v_{2s} - v_{2p_x} + v_{2p_y} - v_{2p_z} \\ & \Phi_4 = v_{2s} - v_{2p_y} - v_{2p_y} + v_{2p_z} \end{split} \tag{9.100}$$
Since the functions $v... | {
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We can assume that three out of the four n = 2 electrons of carbon form sp<sup>2</sup> hybrid orbitals, as in the case of ethylene, the combinations of atomic orbitals being given by [9.101]. A linear combination of one of these functions with the 1s orbital of atomic hydrogen forms a $\sigma$ C—H bond, while the oth... | {
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Transitions between the energy levels of a molecular system can take place with the emission or absorption of radiation. The molecular spectra are more complicated than those of atoms, and in this chapter we shall only analyse some of the simpler features of the rotational, vibrational and electronic spectra. For the m... | {
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Then $I_a = I_b$ and the system forms a symmetrical top, with energy
$$T = \frac{1}{2I_a} (\mathcal{J}_a^2 + \mathcal{J}_b^2) + \frac{1}{2I_c} \mathcal{J}_c^2$$
[10.11]
The component $\mathcal{J}_c$ is along the internuclear line, $\mathcal{J}_c = \mathcal{J}_R$ , and, as we have seen, $\mathcal{J}_R = \Lambd... | {
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Vibrational transitions can occur, due to the interaction with the radiation field, if the matrix element
$$\mathbf{D}_{v'v} = \int \psi_{v'}^{\star} \mathbf{D}(R) \psi_{v} \, dR \qquad [10.17]$$
does not vanish. In this expression $\mathbf{D}(R)$ is the matrix element of the electric dipole moment expressed as a... | {
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In practice, the
Table 10.1 Wave numbers of the central lines in the rotational-vibrational band $(v = 0 \leftrightarrow 1)$ of HCl
| | $\tilde{\nu}$ (cm <sup>-1</sup> ) | A (cm <sup>-1</sup> ) |
|----------------------|-----------------------------------|-----------------------|
| R(5) ... | {
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We note that the Raman effect does not require the existence of a permanent electric dipole moment, but rather than an electric dipole moment should be developed under the influence of the radiation field. For this reason, Raman lines are observed for symmetrical molecules like $H_2$ , $O_2$ , . . . which exhibit no ... | {
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The frequencies of the rotational lines are given by
$$h\nu^{P} = h\nu + B'\mathcal{J}(\mathcal{J} - 1) - B\mathcal{J}(\mathcal{J} + 1)$$
$$h\nu^{Q} = h\nu + B'\mathcal{J}(\mathcal{J} + 1) - B\mathcal{J}(\mathcal{J} + 1)$$
$$h\nu^{R} = h\nu + B'(\mathcal{J} + 1)(\mathcal{J} + 2) - B\mathcal{J}(\mathcal{J} + 1)$$
... | {
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10.11 two electronic absorption transitions leading to the dissociation of a molecule.
It may also happen that an excited state B is coupled by internal perturbations (such as spin—orbit effects) or external ones (such as collisions) to a dissociative state D. In this case the excited state B can either decay to a lo... | {
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In order to obtain the energies of the rotational levels, we notice that the total angular momentum J is obtained by adding to N (the orbital angular
10.14 Angular momentum vectors in Hund's cases (a) and (b).
**(b)**
1... | {
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According to our discussion of Section 2.7, the symmetrical case arises when the nuclei are *fermions*, having half-integer spin (for example $^{16}O_2$ , $^{14}N_2$ ) and the antisymmetrical case occurs when the nuclei are *fermions*, having half-integer spin (for example $^{14}H_2$ , $^{19}F_2$ , $^{127}I_2$ ). ... | {
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Remembering that the electronic ground state of the hydrogen molecule is a $\Sigma_{\mathbf{r}}^+$ state, and that the total function $\Psi_{tot}$ must be antisymmetric in the interchange of the two protons, we see that in this state para hydrogen can only have rotational levels $\mathcal{F} = 0, 2, 4...$ while or... | {
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A pair of wave functions $\hat{\psi}_u$ and $\hat{\psi}_d$ is shown in Fig. 10.18(a) for the case of the lowest (v = 0) vibrational state.
10.18 The wave functions (a) $\hat{\psi}_{\mathbf{u}}$ and $\hat{\psi}_{\mathbf{d}}$ ; (b) $\psi_{\mathbf{u}}$ and $\psi_{\mathbf{d}}$ ; (... | {
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Of course, it should be realised that, in common with all molecular vibrational motions, the inversion spectrum of NH<sub>3</sub> contains a fine and hyperfine structure due respectively to the rotational motion and to magnetic and quadrupole interactions involving the nuclei.
#### **PROBLEMS**
10.1 Show that the r... | {
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Let us consider a typical atomic collision experiment [2] which is illustrated schematically in Fig. 11.1. A homogeneous, well-collimated beam of monoenergetic particles A is directed towards a target containing the scatterers B. We shall assume that the experimental conditions have been chosen in such a way that each ... | {
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Finally, if we denote by $N_{\text{tot}}$ the total number of particles A which have interacted per unit time with target scatterers, we may define a total (complete) cross-section $\sigma_{\text{tot}}$ by the relation
$$N_{\text{tot}} = N_{\text{A}} n_{\text{B}} \sigma_{\text{tot}}$$
[11.11]
If only elastic sc... | {
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The function $\psi_{sc}$ is an asymptotic solution (large r) of the free-particle Schrödinger equation, so that in the large r region the wave function $\psi_{\mathbf{k}_i}(\mathbf{r})$ must satisfy the asymptotic boundary condition
$$\psi_{\mathbf{k}_i}(\mathbf{r}) \underset{r \to \infty}{\sim} A \left[ e^{i\mat... | {
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As shown in Section 2.6, the equations [11.37] can be simplified by introducing the new radial functions
$$u_l(k, r) = rR_l(k, r)$$
[11.38]
which satisfy the equations
$$\left[\frac{d^2}{dr^2} - \frac{l(l+1)}{r^2} - U(r) + k^2\right] u_l(k, r) = 0$$
[11.39]
There is no loss of generality in assuming $u_l(k, r)... | {
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Using the asymptotic formulae [11.47], we see that
$$R_{l}(k, r) = \frac{u_{l}(k, r)}{r}$$
$$\underset{r \to \infty}{\sim} B_{l}(k) \frac{\sin(kr - l\pi/2)}{kr} - C_{l}(k) \frac{\cos(kr - l\pi/2)}{kr}$$
[11.49]
It is convenient to set
$$A_l(k) = [B_l^2(k) + C_l^2(k)]^{1/2}$$
[11.50a]
and
$$\tan \delta_l(k) ... | {
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Indeed, the 'effective (reduced) potential' which occurs in the radial equations [11.37] or [11.39] is
$$U_{\text{eff}}(r) = U(r) + \frac{l(l+1)}{r^2}$$
[11.64]
Thus, as l increases, the centrifugal barrier term $l(l+1)/r^2$ becomes more important and the incident particle needs more energy to overcome this repul... | {
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Since the radial wave function $R_l$ is given in the external region by [11.52], and denoting by
$$\gamma_l = [R_l^{-1}(dR_l/dr)]_{r=a}$$
[11.73]
the value of the logarithmic derivative of the internal solution at r = a, we have
$$\gamma_l(k) = \frac{k[j_l'(ka) - \tan \delta_l(k) n_l'(ka)]}{j_l(ka) - \tan \delt... | {
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Thus
$$\tan \delta_l = \frac{j_l(ka)}{n_l(ka)}$$
[11.94]
and in this case $\gamma_l$ is infinite. Using [11.46] we see that in the low-energy limit $(ka \le 1)$
$$\tan \delta_l \simeq -\frac{(ka)^{2l+1}}{(2l+1)!!(2l-1)!!}$$
[11.95]
so that $|\tan \delta_l|$ quickly decreases as l increases. In fact the low... | {
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In what follows we shall assume for the sake of simplicity that the 'hard sphere' scattering in the lth partial wave (due to $\xi_l$ ) can be completely neglected, together with the contribution of other partial waves to the scattering amplitude [11.58] In that (idealised) case, which corresponds to a pure resonance t... | {
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Let us return to the Schrödinger equation [11.19], which we rewrite as
$$(\nabla^2 + k^2)\psi(k, \mathbf{r}) = U(\mathbf{r})\psi(k, \mathbf{r})$$
[11.114]
where we have indicated explicitly the k dependence of the wave function. The general solution of this equation may be written as
$$\psi(\mathbf{k}, \mathbf{r}... | {
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Returning to [11.124] and remembering that $\mathbf{R} = \mathbf{r} - \mathbf{r}'$ , we therefore obtain a Green's function
$$G_0^{(+)}(k, \mathbf{r}, \mathbf{r}') = -\frac{1}{4\pi} \frac{e^{ik|\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|}$$
[11.126]
which exhibits the required purely outgoing wave behaviour... | {
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We obtain in
this way the sequence of functions
$$\psi_{0}(\mathbf{r}) = \Phi_{\mathbf{k}_{i}}(\mathbf{r}) = e^{i\mathbf{k}_{i}\cdot\mathbf{r}}$$
[11.136a]
$$\psi_{1}(\mathbf{r}) = \Phi_{\mathbf{k}_{i}}(\mathbf{r}) + \int G_{0}^{(+)}(\mathbf{k}, \mathbf{r}, \mathbf{r}')U(\mathbf{r}')\psi_{0}(\mathbf{r}') d\mathbf{r... | {
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It is interesting to note from [11.146] that
$$\lim_{k \to \infty} \left[ k^2 \sigma_{\text{tot}}^{\text{Bl}}(k) \right] = 2\pi \int_0^\infty |f_{\text{Bl}}(\Delta)|^2 \Delta \, d\Delta \qquad [11.147]$$
Since the right-hand side of [11.147] is independent of k, we see that $\sigma_{\text{tot}}^{\text{B1}}$ is pr... | {
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Until now we have considered the scattering of a spinless particle by a *real* potential $V(\mathbf{r})$ , a problem which is equivalent to the elastic scattering between two structureless particles. However, we have seen in Section 11.1 that when a particle collides with a target, non-elastic scattering may occur. Th... | {
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Using the methods of Appendix A2, show that the centre of mass and laboratory differential cross-sections for observation of the particle C in a given direction are related by
$$\frac{\mathrm{d}\sigma_{\mathrm{C}}}{\mathrm{d}\Omega_{\mathrm{I}}}\left(\theta_{\mathrm{L}},\,\phi_{\mathrm{L}}\right) = \frac{\left(1\,+\,... | {
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With the help of the results obtained in the last chapter, for scattering of a beam of particles by a potential, we are now ready to discuss electron scattering by atoms, or by ions. In order to explain how cross-sections for elastic and inelastic scattering can be calculated, we shall, for the most part, take the simp... | {
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If the incident energy is insufficient to excite the hydrogen atom to the n=2 level, the wave function, for $r_1 \gg r_2$ , must represent electron 1 moving with respect to a ground state hydrogen atom containing electron 2, and we have
$$\psi_{\pm}(\mathbf{r}_1, \mathbf{r}_2) \sim F_1^{\pm}(\mathbf{r}_1)\psi_{100}(... | {
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To obtain an equation for the function $F_1^{\pm}(\mathbf{r}_1)$ , we notice that the projections of the Schrödinger equation [12.1] onto the complete set of
orthonormal hydrogenic functions $\psi_q$ must all vanish:
$$\int \psi_q^{\star}(\mathbf{r}_2) \left[ -\frac{1}{2} \nabla_1^2 - \frac{1}{2} \nabla_2^2 - \f... | {
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For any atom, the potential corresponding to [12.19] can be written down by recognising that $|\psi_1(\mathbf{r}_2)|^2$ is the electron density $\rho(\mathbf{r}_2)$ , and in general, for an atom with nuclear charge Z
$$V_{11}(\mathbf{r}_1) = -\frac{Z}{r_1} + \int \frac{\rho(\mathbf{r}_2)}{|\mathbf{r}_1 - \mathbf{r... | {
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It is found that
$$(\nabla_1^2 + k_q^2) F_q^{\pm}(\mathbf{r}_1) = 2 \sum_{q'=1}^N V_{qq'}(\mathbf{r}_1) F_{q'}^{\pm}(\mathbf{r}_1) \pm 2 \sum_{q'=1}^N \int K_{qq'}(\mathbf{r}_1, \mathbf{r}_2) F_{q'}^{\pm}(\mathbf{r}_2) d\mathbf{r}_2$$
$$q = 1, 2, \dots N \quad [12.38]$$
These are a set of N coupled equations whic... | {
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Since the exchange part of $V_{\text{opt}}^{\pm}$ is non-local, the explicit form of [12.41] reads
$$[\nabla^{2} + k_{1}^{2} - 2V_{11}(\mathbf{r}_{1}) - 2V_{pol}(\mathbf{r}_{1})]F_{1}^{\pm}(\mathbf{r}_{1})$$
$$= \pm 2 \int K_{11}(\mathbf{r}_{1}, \mathbf{r}_{2})F_{1}^{\pm}(\mathbf{r}_{2}) d\mathbf{r}_{2}$$
[12.49]... | {
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Hence the lowest order channel functions $F_{q}^{\pm}$ are given by
$$F_q^{\pm}(\mathbf{r}_1) \equiv F_q(\mathbf{r}_1) = \delta_{q1} \exp(i\mathbf{k}_i \cdot \mathbf{r}_1)$$
[12.56]
Inserting this lowest order approximation into the right-hand side of [12.38],
and neglecting the exchange potentials, we have
$... | {
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When the change of energy of the target atom $(E_q - E_1)$ is small compared with the incident energy $(k_1^2/2)$ , we can write
$$\Delta_{\text{max}} \simeq 2k_1$$
and $\Delta_{\text{min}} \simeq (E_q - E_1)/k_1$ [12.70]
In order to see the connection between cross-sections for transitions induced by electron... | {
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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If the dipole term vanishes, as in $s \to s$ or $s \to d$ transitions, we see from [12.73], [12.79] and [12.80] that the high energy cross-section decreases like $E^{-1}$ , and
$$\sigma_{\bar{q}1}(k_1) \simeq \frac{\pi}{k_1^2} \sum_{m} \left| \left\langle \psi_q \left| \sum_{i=1}^Z z_i^2 \right| \psi_1 \right\... | {
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Because of the finite lifetime of this level, the energy $E_{\rm r}$ of the level is not sharp, but has a width, in this case $\sim 0.17$ eV.
Although originally doubly excited levels were discovered spectroscopically by observing emission lines, which because of the competitive radiationless decay are weak and b... | {
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The configurations that are mixed are all of the form 2snl and 2pnl. In the absence of the open channel, the equations [12.38] are eigenvalue equations and possess bounded solutions for $F_2^{\pm}$ and $F_{3m}^{\pm}$ only at discrete energies, that is for discrete negative values of $k_q^2$ . (Notice that because ... | {
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In Chapters 9 and 10, we discussed in some detail the properties of diatomic molecules, which are the bound states of two atoms or ions. We shall now turn to the case in which one atom is scattered by another. Such processes occur naturally in an assembly of atoms, such as in a gas, but can also be studied experimental... | {
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Then, as we are taking the Z direction to be along AB,
$$r_{12} = [(x_{2B} - x_{1A})^2 + (y_{2B} - y_{1A})^2 + (z_{2B} - z_{1A} + R)^2]^{1/2}$$
$$r_{1B} = [x_{1A}^2 + y_{1A}^2 + (z_{1A} - R)^2]^{1/2}$$
$$r_{2A} = [x_{2B}^2 + y_{2B}^2 + (z_{2B} + R)^2]^{1/2}$$
[13.5]
Each of the terms $1/r_{12}$ , $1/r_{1B}$ ... | {
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One of the most widely used of these is the Lennard-Jones potential, which has the form
$V(R) = C \left[ \frac{1}{2} \left( \frac{R_0}{R} \right)^{12} - \left( \frac{R_0}{R} \right)^6 \right]$ [13.16]
where C and $R_0$ are constants. The constant C can be related to $C_W$ (the van der Waals constant), but both... | {
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Except at very small angles, and apart from some small interference effects that we will discuss briefly later, the considerations of the previous paragraph show that the elastic scattering of one atom by another can be determined by purely Newtonian mechanics. We have derived in Appendix 1 the classical cross-section ... | {
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"Header 3": "13.3 THE ELASTIC SCATTERING OF ATOMS AT LOW VELOCITIES",
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On the other hand for classical conditions to apply we see from [13.25] that we must also have $\theta \gg \theta_c$ , where the critical angle $\theta_c$ is of the order of a few milliradians.
Experiments can be analysed to determine s and A by plotting $\log [\theta \sin \theta d\sigma/d\Omega]$ against $\log... | {
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"Header 3": "13.3 THE ELASTIC SCATTERING OF ATOMS AT LOW VELOCITIES",
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In this section we will discuss the processes in which an atom is excited during a collision, or in which an electron is transferred from one atom to another. We will treat the case in which the kinetic energy of the relative motion of two atoms is very large compared with the change in electronic energy of either atom... | {
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"Header 3": "13.4 ELECTRONIC EXCITATION AND CHARGE EXCHANGE",
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A free electron with this momentum and energy would have the wave function
$$\psi = e^{i\mathbf{p}\cdot\mathbf{r}} e^{-iEt}$$
[13.55]
The proper forms of the unperturbed wave function at large |t| are therefore given by
$$\Psi \underset{t \to -\infty}{\sim} \psi_{1s}(r_{\rm B}) {\rm e}^{-iE_{1s}t} {\rm e}^{-i{\bf... | {
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"Header 3": "13.4 ELECTRONIC EXCITATION AND CHARGE EXCHANGE",
"token_count": 1984,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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To satisfy the boundary conditions $a(b, t = -\infty) = 1$ and $c(b, t = -\infty) = 0$
To satisfy the boundary conditions $a(b, t = -\infty) = 1$ and $c(b, t = -\infty) = 0$ we must take from [13.60]
$$A^{\pm}(b, t = -\infty) = \pm \frac{1}{\sqrt{2}}$$
[13.67]
The solutions of [13.65] satisfying these cond... | {
"Header 1": "13 Atom-atom collisions",
"Header 3": "13.4 ELECTRONIC EXCITATION AND CHARGE EXCHANGE",
"token_count": 2009,
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This is possible because the Schrödinger equation for an electron moving in the field of two identical charges Z, is (in atomic units)
$$\left[ -\frac{1}{2} \nabla^2 - \frac{Z}{|\mathbf{r} - \mathbf{R}/2|} - \frac{Z}{|\mathbf{r} + \mathbf{R}/2|} \right] \Psi(\mathbf{r}, t) = i \frac{\partial}{\partial t} \Psi(\mathbf... | {
"Header 1": "13 Atom-atom collisions",
"Header 3": "13.4 ELECTRONIC EXCITATION AND CHARGE EXCHANGE",
"token_count": 1933,
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} |
The coupled equations [13.85] can then be approximated by
$$i\dot{a}_{k}(b, t) = V_{ki}(t)e^{i(E_{k}-E_{i})t}$$
[13.88]
with the solution (for $k \neq i$ )
$$a_k(b, t) = -i \int_{-\infty}^{t} V_{ki}(t') e^{i(E_k - E_i)t'} dt'$$
[13.89]
The probability of finding the system after the collision in the state k is... | {
"Header 1": "13 Atom-atom collisions",
"Header 3": "13.4 ELECTRONIC EXCITATION AND CHARGE EXCHANGE",
"token_count": 2034,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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The overall angular momentum of an atom, arising from both the orbital and the spin angular momenta of the electrons, can either be zero as for closed shell atoms, or non-zero, as for the ground state of atomic hydrogen $(j = s = \frac{1}{2}; l = 0)$ . When the total angular momentum **J** is non-zero, an atom possess... | {
"Header 1": "14.1 MAGNETIC RESONANCE AND THE MEASUREMENT OF THE GYROMAGNETIC RATIOS",
"token_count": 2034,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
Dropping these terms, we find the approximate equations:
$$i\dot{A}_{+} = \frac{1}{4}\bar{\omega}_{0} \exp[i(\omega_{0} - \omega)t] A_{-}$$
$$i\dot{A}_{-} = \frac{1}{4}\bar{\omega}_{0} \exp[-i(\omega_{0} - \omega)t] A_{+}$$
[14.15]
#### **Exact resonance**
It is easy to verify that in the case of exact resonanc... | {
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"token_count": 2027,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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