Particle – Wave Duality
In 1924, de Broglie proposed that matter exhibits wave-particle duality, similar to light, which can behave as both a wave and a particle He established that matter's wavelength follows the equation λ = h/p, where p represents linear momentum, paralleling the relationship for light By applying Einstein's equations, he derived that λ = h/mc for relativistic mass, acknowledging that a photon has zero rest mass For a particle with velocity v, momentum is given by p = mv, leading to λ = mv/h Notably, the phase velocity of the de Broglie wave associated with a particle is vp = λc^2/v, indicating that vp exceeds the speed of light (c), suggesting that the de Broglie wave travels faster than the particle itself, which is counterintuitive This led to the understanding that a particle is better represented by a wave packet—a collection of waves that interfere constructively in a localized region, allowing for the particle's presence, while destructively interfering elsewhere to diminish amplitude rapidly.
Simplest type of a wave is a plane monochromatic wave
Jðr;tị ẳe ẵiðk$r utị (1.2.1)
Using relationsEẳh – uandpẳh – k, for the energy and linear momentum, this equa- tion may also be written as,
Jðr;tị ẳeẵZ i ðp$rEtị (1.2.2) whereuandkare the angular frequency and wave vector of the plane wave, respectively.
A wave packet is constructed by superposition of waves by Fourier relation Thus, for one spatial dimension the wave packet is
Aðkịe ẵiðkxutị dk (1.2.3) or in terms of energy and linear momentum,
The behavior of such a wave group in time is determined by the way in which the angular frequencyu(ẳ2pn) depends upon the wave numberk ẳ 2p l
, i.e., by the law of dispersion.
A wave packet travels at a specific speed known as the group velocity (vg), which corresponds to the particle's velocity This connection between wave packets and particles helps to elucidate Heisenberg’s uncertainty principle In 1927, the experiments by Davisson and Germer, which involved bombarding metals with electrons, provided evidence for de Broglie’s theory through the observation of diffraction patterns.
Matrix Mechanics and Wave Mechanics
In late 1925, Werner Heisenberg, alongside Max Born and Pascual Jordan, introduced matrix mechanics, a groundbreaking theory that substitutes traditional physical quantities such as particle coordinates, momenta, and energies with matrices This innovative approach established rules for matrix manipulation, enabling predictions that could be tested against experimental results.
Erwin Schrödinger developed a theory based on the Hamilton–Jacobi formulation of classical mechanics, which describes particle behavior through a wave equation This approach led to the formulation of the Schrödinger equation, which was experimentally validated In Schrödinger's wave mechanics, systems like atoms or molecules are represented by a wavefunction, making the theory more accessible and visually intuitive compared to matrix mechanics Its foundation on established classical mechanics contributed to its widespread acceptance in the scientific community.
Matrix mechanics and wave mechanics predict exactly the same results for experi- ments This suggests that they are really different forms of a more general theory In
In 1930, Paul Dirac introduced a comprehensive formulation of quantum mechanics that remains in use today This framework allows for the derivation of both matrix mechanics, known as the Heisenberg picture, and wave mechanics, referred to as the Schrödinger picture Additional representations can also be derived, and the choice of which to use in practice depends on convenience In the field of quantum chemistry, the Schrödinger picture is typically the more straightforward option.
Relativistic Quantum Mechanics
Non-relativistic quantum mechanics (non-RQM) formulates quantum mechanics within Galilean relativity by quantizing classical mechanics through operator substitution, while relativistic quantum mechanics (RQM) integrates special relativity into quantum theory RQM has proven more effective in various applications, such as predicting antimatter, electron spin, and the behavior of charged particles in electromagnetic fields In chemistry, relativistic effects act as minor corrections to the nonrelativistic theory derived from the Schrödinger equation, influencing electron behavior based on their speeds relative to light These effects are particularly significant in heavy elements, where electrons reach relativistic speeds Quantum chemistry employs RQM to elucidate the properties and structures of heavy metals, notably explaining the unique color of gold, which diverges from the typical silvery appearance of most metals due to relativistic influences.
Schrửdinger Wave Equation
Time-Independent Schrửdinger Wave Equation
When the Hamiltonian is independent of time the general solution (J) of the Schro¨dingerEq (1.5.1)can be expressed as a product of function of spatial position and time Thus
By substituting the equation J(x,t) = F(x)f(t) into the time-dependent Schrödinger equation, we derive the time-independent Schrödinger equation for one dimension: -ħ²/(2m) d²F(x)/dx² + V(x)F(x) = EF(x) Additionally, the time component is expressed as f(t) = Ce^(iEt), where C is a constant.
The total wavefunction is therefore
Jðx;tị ẳFðxị$e iEt – h (1.5.10) Equation (1.5.8)can also be written as
The eigenvalue equation, represented as Equation (1.5.11), defines the energy eigenvalue E, indicating that a state with a specific energy has a corresponding wavefunction as described in Equation (1.5.10) Additionally, Equation (1.5.1) establishes a continuity equation within the framework of quantum mechanics.
The equation presented parallels the continuity equation found in electrodynamics, where \( r \) represents charge density and \( J \) denotes current density By comparing equations (1.5.3) and (1.5.4), we derive that \( r = J = J = F = F = |F|^2 \), which is referred to as the position probability density.
2miðJ VJJVJ ị (1.5.16) by analogy toEq (1.5.14)is called the probability current density.
From the above, it follows that if div JẳV$Jẳ0; the probability density r will be a constant in time Such states are called stationary states and are independent of time.
J Jdefines the probability of finding a particle in unit volume element Since the probability of finding the particle somewhere in the region must be unity,
J ðr;tịJðr;tịdsẳ1 (1.5.17) wheredsis the three-dimensional volume elementdx dy dz The functionJis now said to be normalized and the above equation is said to be the normalization condition.
Schrửdinger Equation in Three-Dimensions
In a three-dimensional space the wave packet can be written as
Aðkịexp ẵiðk$rutị dk (1.5.18) and the time-dependent Schro¨dingerEq (1.5.1)is replaced by h – 2 2mV 2 Jðr;tị ỵVðrịJðr;tị ẳih – vJðr;tị vt (1.5.19) where,
V 2 ẳ v 2 vx 2 þ v 2 vy 2 þ v 2 vz 2 (1.5.20) is known as Laplacian operator The time-independent Schro¨dinger equation is written as
For spherically symmetric physical systems, utilizing spherical polar coordinates to express the Schrödinger equation is advantageous In three-dimensional scenarios, the Laplacian in spherical polar coordinates simplifies the formulation of the Schrödinger equation.
Expanded, it takes the form h – 2
1 r 2 sinq sinq v vr r 2 v vr þ v vq sinq v vq þ 1 sinq v 2 vf 2
Jðr;q;fị ỵVðrịJðr;q;fị ẳEJðr;q;fị or h – 2
1 r 2 v vr r 2 v vr þ 1 r 2 sinq v vq sinq v vq þ 1 r 2 sin 2 q v 2 vf 2
Jðr;q;fị ỵVðrịJðr;q;fị ẳEJðr;q;fị
(1.5.23)This is the form best suited for the study of the hydrogen atom.
Operators—General Properties, Eigenvalues, and Expectation Values
Some Operators in Quantum Mechanics
In classical mechanics, dynamical variables are typically represented using position and momentum coordinates such as x, y, z, px, py, and pz In contrast, quantum mechanics involves constructing operators by taking classical expressions and substituting position coordinates and linear momenta with their corresponding quantum operators.
Operators corresponding to any function of position, such as x, or potential V(x) are simply the functions themselves.
As shown above, linear momentum is represented in the operator form as b p x ẳ i – hv vx ðsimilarly for theyandzcomponentsị:
The time-dependent Hamiltonian is represented as Hbðtị ẳi – h vt v , while in the time- independent form, if may be written as
Hbẳ p 2 x 2mþ p 2 y 2mþ p 2 z 2mỵVðrị ẳh – 2
2m v 2 vx 2 þ v 2 vy 2 þ v 2 vz 2 ỵVðrị ẳ h – 2 2mV 2 ỵVðrị
For a classical particle with linear momentum p and position vector r, the orbital angular momentumLis:
Lbẳrpbẳ i – hrV (1.6.5) or in Cartesian components,
Lbxẳypb z zpb y ẳ ih – y v vzz v vy
Without going into mathematical rigors it may be mentioned that in terms of spherical polar coordinates
Lb y ẳ ih – cosf v vqcotqsinf v vf
Lb 2 ẳ h – 2 1 sinq v vq sinq v vq þ 1 sin 2 q v 2 vf 2
Properties of Operators
1 Two operators AbandBbare said to be equal if Afb ẳBf for all functions f.b
2 Product of two operators is defined by the equation AbBfðxị ẳb AẵbBfðxị:b
3 Operators obey the associative law of multiplication Aðb BbCbị ẳ ðAbBịb C:b
4 Unlike ordinary algebra, the operators do not obey the commutative law of multi- plication While,abẳbain ordinary algebra, AbBband BbAbare not necessarily equal operators.
CommutatorẵA;b Bb of operators Aband Bbis defined as bA;Bb ẳAbBbBbAb
In the context of operators A and B, they are considered to commute if applying them in either order yields the same result, specifically when the expression A;B equals 0 Conversely, if A;B does not equal 0, the operators do not commute For instance, when A is defined as a function of x and B as a function of v, the relationship between the operators can be expressed as A;B f(x) = A B f(x) = x v v x f(x) = f(0).
Simultaneous measurements of physical quantities are only certain when represented by commuting operators; otherwise, uncertainty arises This uncertainty in measurements is quantified by Heisenberg's uncertainty principle.
5 Eigenfunctions of commuting linear operators are simultaneous eigenfunctions or in the case of degeneracy can be constructed by superposition principle to be simultaneous.
An eigenfunction is said to be a simultaneous eigenfunction of two linear opera- torsAband Bbfor eigenvaluesaandb, respectively, if it satisfies the condition
BJb ẳbJ Conversely, two operators shall commute if they have simultaneous eigenfunctions. Operators having simultaneous eigenfunctions are said to be compatible.
6 Some other properties of commutators are: bA;Bb ẳ bB;Ab kA;b Bb ẳ bA;kBb ẳk bA;Bb bA;BbþCb ẳ bA;Bb þ bA;Cb and bAþB;b Cb ẳ bA;Cb þ bB;Cb bA;BbCb ẳ bA;Bb bCỵBb bA;Cb and bAB;b Cb ẳ bA;Cb bBỵAb bB;Cb
(1.6.9) wherekis a constant and the operators are assumed to be linear.
1.6.2.1 Commutation Properties of Linear and Angular Momentum Operators Thex,y, andzcomponents of the linear momentum operators commute with each other but not with the corresponding position coordinates. pb x ;pb y ẳ b p y ;pb z ẳ b p z ;pb x ẳ0 pb x ;x ẳ b p y ;y ẳ b p z ;z ẳ ih –
The physical significance of these relations follows from Heisenberg’s uncertainty principle according to which ifDA andDB are the uncertainty of measurement of two dynamical variables represented by operatorsAband Bbthen
While the components of momentum can be measured with certainty, there remains inherent uncertainty in accurately determining a particle's position and its associated momentum component.
The commutation relations for the angular momentum operators are: bLx;Lby ẳi – hLbz; bL z ;Lb x ẳi – hLb y ; bLy;Lbz ẳih – Lbx; and
The x, y, and z components of angular momentum operators do not commute; however, the square of the angular momentum operator, L², commutes with Lx, Ly, and Lz Consequently, the magnitude of the angular momentum vector can be accurately measured alongside its z-component.
Postulates of Quantum Mechanics
Heisenberg made some postulates in the development of quantum mechanics. These are:
In quantum mechanics, the state of a system is described by a mathematical function known as the wavefunction, which encapsulates all information about a particle in a conservative force field This wavefunction must be single-valued with respect to both position and time, ensuring a clear and unambiguous probability of locating the particle at any given moment and location.
With every physical observableqthere is associated an operatorQ It turns out that theb operators occurring in quantum mechanics are linear operators.
The only possible values which a single measurement of an observable associated with an operator can yield are the eigenvaluesq of the equation
Eigenvalues of the operator Qb correspond to the eigenfunction J, indicating that the equation can be satisfied by multiple wavefunctions, often infinitely many, known as eigenfunctions and eigenvalues There are two potential scenarios that may arise in this context.
1 There is only one independent eigenfunction for one eigenvalue In this case the physical state is said to be nondegenerate.
2 There are a number of eigenfunctions for a given eigenvalue In this case, the state is said to be degenerate.
Any operatorQb associated with a physically measurable propertyqwill be Hermitian.
An operator is said to be Hermitian if for any two functionsJandVit satisfies the condition
The asterisk as superscript indicates the complex conjugate of the quantity imme- diately to its left Hermitian operators are linear operators and have real eigenvalues.
The eigenfunctions of an operator Q constitute a complete set of linearly independent functions In cases where the eigenvalue q of the operator Q is degenerate, any linear combination of these independent eigenfunctions remains an eigenfunction.
This is known as the superposition principle.
If J1, J2, Jn are a complete set of linearly independent eigenfunctions of the operatorQrepresenting a physical system, then these functions are said to be normal if
These two equations combined together may be written as
J i Jjdsẳdij; (1.7.4) wheredijẳ1; ifiẳj and ẳ0; ifisj
Equation (1.7.4)represents the orthonormality conditions of the eigenfunctions.
This principle connects quantum mechanical measurements with classical ones, asserting that the "expectation value" calculated in quantum mechanics corresponds to the average value obtained from numerous physical measurements.
According to this postulate, if a system at a given timetis characterized by normalized wavefunctionJof the observable QbthenR
J QJdsb gives the average of a large number of measurements performed on this system under the same initial physical conditions.R
J QJdsb is known as the expectation value ofQbin the state Jat timet. hqi ẳ
The meaning of the expectation value in terms of the classical measurements can be understood in the following manner:
It is a practice to repeat uncertain experiments several times and to determine the average of several measurements on a chosen parameter.
If a certain valuexkfor the parameterxoccursnktimes such thatPm kẳ1nkẳnthen the average value ofxmay be written as xẳ
IfQ k is the probability that a measurement yields a valuex k , then
FromEqs (1.7.6) and (1.7.7), we get xẳX m kẳ1
In quantum mechanics, the average value x is called the expectation value and is written ashxi:
If the distribution of measurements is continuous then xẳ
N x dW (1.7.9) wheredWẳQ(x)dxis the probability that xhas a value betweenxandxỵdx.
Since the total probability for x to have some value is 1, Q(x) must satisfy the condition
Q(x) is called the probability density of x because it gives the probability per unit interval.
The weighted average of x, as derived from Eq (1.7.9), emphasizes the contribution of more probable values of x, which play a significant role in the integral calculation.
The above procedure can be used to get expectation value of a function of one or several parameters. hfðx;y;zịi ẳ
Z Z Z Qðx;y;zịfðx;y;zịdx dy dz (1.7.12)
In quantum mechanics, the expectation value is crucial for linking theoretical calculations to observable laboratory measurements Specifically, for the position \( x \), the expectation value is mathematically represented as \( \langle x \rangle \).
This integral can be interpreted as the average value ofx that we would expect to obtain from a large number of measurements.
Hydrogen Atom
Solution of Schrửdinger Equation for Hydrogen-Like Atoms
The hydrogen problem is effectively addressed using spherical polar coordinates In this context, let \( Z e \) represent the charge of the nucleus with mass \( m_N \) in a hydrogen-like atom, while \( e \) denotes the charge of the electron with mass \( m_e \) The coordinates \( (x, y, z) \) describe the electron's position relative to the nucleus, and \( r \) signifies the distance between the two particles According to Coulomb's law, the attractive force acting between these charged particles is directed along the line connecting them and is characterized by a specific magnitude.
The potential energy resulting from this force is
The central force field two-body problem can be simplified into a one-body problem by utilizing the concept of reduced mass, denoted as m, which is calculated using the formula m = (m_e * m_N) / (m_e + m_N), where m_e represents the mass of one body and m_N represents the mass of the other body.
On replacingmby mand usingEq (1.8.2), the Schro¨dingerEq (1.5.23) in spherical polar coordinates can therefore be written as
1 r 2 v vr r 2 vJ vr þ 1 r 2 sinq v vq sinqvJ vq þ 1 r 2 sinq v 2 J v 2 f þ2m
Jẳ0 (1.8.3) whereJis a function ofr,q, andf.
This equation can be separated into the radial and angular parts by writing
Jðr;q;fị ẳRðrị$Yðq;fị (1.8.4)
SubstitutingEq (1.8.4)into Eq (1.8.3), we get
FIGURE 1.1 Hydrogen-like atom in spherical polar coordinates. or
The two sides of the equation, being functions of an independent variable, must equal a constant, denoted as A This leads to the formulation of two distinct equations: one that is based on the radial coordinate (r) and another that relies on spherical coordinates (q, f) The resulting equations can be expressed as v vr r² vR vr + 2mr².
1 sinq v vq sinqvY vq þ 1 sin 2 q v 2 Y vf 2 ỵAYẳ0 (1.8.8)
Equation (1.8.8)can also be separated into a polar part (q) and an azimuthal part (f) by writing
Yðq;fị ẳQðqịFðfị (1.8.9) SubstitutingEq (1.8.9)inEq (1.8.8)and multiplying by sin QF 2 q ; sinq Q v vq sinqvQ vq þ1 F v 2 F vf 2 ỵAsin 2 qẳ0 (1.8.10) or sinq
Q v vq sinqvQ vq ỵAsin 2 qẳ 1
Since the two sides ofEq (1.8.7) are functions of different variables, they must be equal to some constant, saym 2
Thus the Schro¨dingerEq (1.8.3) breaks into three equations—the radial Eq (1.8.7), the angularEq (1.8.12), and the azimuthalEq (1.8.13).
The normalized solution to the azimuthalEq (1.8.13)is simple and may be written as
The magnetic quantum number \( m \) can only take integer values \( m = 0, 1, 2, \ldots \), which defines the orientation of the orbital angular momentum \( L \) in relation to an external coordinate system While this quantum number does not influence the energy of an isolated one-electron atom, it becomes significant in the presence of an external magnetic field, leading to varying energy levels based on the orientation of the orbital angular momentum In the absence of external influences, orbitals with the same \( m \) value do not have a preferred orientation The functions \( F_m(m) \) can be expressed in both complex and real forms, with either representation being acceptable, as demonstrated in Table 1.1.
The formal solutions toEq (1.8.12)can be found in terms of the associated Legendre polynomial The expression for the solutions ofQfunction after normalization is
The associated Legendre Polynomial of degree \( l \) and order \( m \) is represented as an infinite series expansion These polynomials serve as solutions to a specific category of second-order differential equations, referred to as Legendre's equation.
For the series expansion to terminate, the constant A must equal l(l + 1), where l is an integer ranging from 0 to (n - 1) and represents the orbital quantum number of an electron around the nucleus The angular momentum is given by the formula h√(l(l + 1)), making l also known as the angular momentum quantum number In one-electron systems without magnetic fields, l does not contribute to the overall energy However, when electric charges exhibit net angular momentum, they generate a magnetic moment The presence of an external magnetic field or a magnetic moment from the nucleus will interact with the orbital magnetic moment, leading to energy variations This interaction causes the energy levels of orbitals with different l values to split in polyelectron systems.
ffiffiffiffiffiffi p 2p e if or F 1 cos ðfị ẳ 1
ffiffiffiffiffiffi p 2p e ði2fị or F 2 sin ðfị ẳ 1
ffiffiffi p p sin 2f atoms Some of theQlm(q) functions corresponding tolẳ0 (sorbitals),lẳ1 (porbitals) andlẳ2 (dorbitals) frequently used in quantum chemistry of organic systems are given inTable 1.2.
1.8.1.3 Solution of the Radial Equation
An analysis of the radial component of the Schrödinger Equation reveals its limiting behavior for large values of r (r/N) In this scenario, the equation simplifies to d²R/dr² = (2mE).
The most general solution to this equation is an exponential function:
An acceptable wavefunction must be finite at all points, which restricts us to using only negative exponents As a result, we observe that our wavefunction exhibits exponential decay at large values of r.
For small values ofrthe radial function can be assumed to have some form such as
F(r) being simply some arbitrary function of r If F(r) is evaluated it takes the form of what is known as the associated Laguerre polynomial which again is an infinite series
4 sin 2 q expansion To terminate the series an integernmust be introduced The final form of the radial function after normalization is given inEq (1.8.20).
4p 2 me 2 (1.8.21) rẳ 2Z na0 r (1.8.22) a0is known as Bohr’s radius having value 0.52918A.
L 2lþ1 nþl is the associated Laguerre polynomial of degree {(nþl)(2lþ1)} and order (2lþ1).
For a satisfactory solution of the radial Eq (1.8.20), the degree of the Laguerre Polynomial must be a positive integer So, (nỵl)(2lỵ1)ẳ(nl1)0 ornlỵ1. Sincelmay have values 0, 1, 2,.;nmay have values 1, 2, 3,.N.
The energy for a hydrogenic system is
The principal quantum number, denoted as n, is the key quantum number that determines the energy levels of one-electron atoms in the absence of external influences It is primarily associated with the radial component of the wavefunction, indicating that it significantly influences the energy of the hydrogen atom This relationship is logical, as the potential energy is determined solely by the distance of the electron from the nucleus, represented by the radial coordinate r Notably, the most probable electron distributions for different values of n correspond to the radii of the Bohr orbits with matching energy levels.
Some of the hydrogen-like radial wavefunctions fornẳ1 (K shell),nẳ2 (L shell), and nẳ3 (M shell) are given inTable 1.3.
The Charge-Cloud Interpretation of J
The wavefunction can be interpreted probabilistically, where J² ds represents the likelihood of finding an electron within a volume ds Alternatively, a more visual interpretation likens the electron to a charge cloud, with its density at any point proportional to J²; thus, a higher J² indicates a denser charge cloud and a greater concentration of negative charge This approach shifts the focus from probability density to actual particle density However, while the charge-cloud model is a useful visualization, it is not entirely accurate, as a single electron, being a particle, cannot be spread across regions the size of an atom or molecule.
Coulson (1961) demonstrated that by conducting numerous measurements of an electron's position and representing each position as a tiny dot in three-dimensional space, we can create a visual representation akin to a cloud When these dots are indistinguishable, the resulting diagram resembles a charge cloud, with denser areas indicating a higher probability of locating the electron This density serves as a direct measure of the probability function, illustrating the likelihood of finding the electron in various locations Although the concept of a charge cloud may lack strict validity, it proves to be a valuable tool Additionally, electron wavefunctions extend infinitely from the nucleus, indicating a finite probability of finding the electron even at considerable distances, with approximately 90-95% of the charge contained within a specific contour.
Table 1.3 Hydrogen-like radial wavefunctions R nl (r) n [ 1, K shell l ẳ 0 ; 1s R 10 ðrị ẳ ðZ = a 0 ị 3 = 2 $ 2e r 2 n [ 2, L shell l ẳ 0 ; 2s R 20 ðrị ẳ ðZ = a 0 ị 3=2
2 ffiffiffi p 6 $ re r 2 n [ 3, M shell l ẳ 0 ; 3s R 30 ðrị ẳ ðZ = a 0 ị 3=2
The charge cloud concept illustrates the distribution of electric charge in space, influencing the stereochemical arrangement of atoms in polyatomic systems This understanding is essential for analyzing the ground state of the hydrogen atom.
Normal State of the Hydrogen Atom
The total wavefunctionJ(r,q,f) for the hydrogen atom in its ground statenẳ1,lẳ0, andmẳ0 can be obtained by usingTables 1.1–1.3andEqs (1.8.14), (1.8.16), and (1.8.20). Since,
Jnlmðr;q;fị ẳR nl ðrịQlnðqịFmðfị (1.8.24) we get in this case,
From this it follows that rẳJ Jẳ 1 pa 3 0 e 2r=a 0 (1.8.27)
The values of J and J² are illustrated in various forms in Figure 1.2 Given the spherical symmetry of the atom, the radial distance r is the sole variable Rather than displaying the density ρ(r), we represent the radial density as 4πr²ρ(r), where 4πr²dr denotes the volume element between the spheres of radius r and r + dr Consequently, 4πr²ρ(r)dr reflects the total probability of finding the electron at a distance between r and r + dr from the origin Figure 1.3 depicts this radial density, revealing that the maximum radial density occurs at r = a₀, consistent with Bohr's classical theory; however, we now refer to the locus of points representing maximum probability density instead of an orbit.
FIGURE 1.2 (a) Charge cloud, (b) boundary surface, and (c) variation of probability function j J j 2 in space for hydrogen atom.
The radial distribution of hydrogen-like atoms varies with different quantum states, as illustrated in Figure 1.3(b) For higher quantum numbers (n), the curve intersects the radial axis (nl1) a specific number of times between r=0 and r=N, indicating nodal points In this context, the normal 1s state of hydrogen can be visualized as a concentric spherical ball surrounding the nucleus In contrast, the 2s state consists of a central ball with an outer shell, the 3s state features a ball with two concentric shells, and the 6s state comprises a ball with five concentric shells, as depicted in Figure 1.4.
Atomic Orbitals
The wavefunction Jnlm describes electron motion and allows for the calculation of the probability of locating an electron within a specific area around an atom's nucleus, thus defining it as an atomic orbital In spectroscopy, the orbital quantum number (l) is represented by the symbols s, p, d, f, and g, corresponding to values of l = 0, 1, 2, 3, etc Typically, atomic orbitals are represented using spherical coordinates (r, θ, φ) for atoms and Cartesian coordinates (x, y, z) for polyatomic molecules.
Atomic orbitals, such as those of the hydrogen atom, are often illustrated using simple representations of their shapes, which encompass 90–95% of the electron probability density To depict the phases of these orbitals, the functions J(r, q, f) are sometimes plotted alongside the more commonly used |J(r, q, f)|² functions While the graphs of J(r, q, f) and |J(r, q, f)|² are quite similar, they serve different purposes in visualizing the properties of atomic orbitals.
FIGURE 1.3 Radial distribution function or density for the ground state of (a) hydrogen and (b) hydrogen-like atoms.
Atomic orbitals exhibit distinct shapes, with certain orbitals displaying less spherical and thinner lobes The shape of an orbital is influenced by the quantum number l, while its orientation is determined by the quantum number ml Additionally, since some orbitals are defined using complex number equations, their shapes can also be affected by the value of ml.
Following Mulliken, we occasionally refer to, one-electron orbital wavefunctions, such as the hydrogen-like wavefunctions, as orbital and use the function
Y m l l ðq;fị ẳQl;mðqịVmðfị (1.9.1) to determine the shape of the orbital.
Thus, fornẳ2, we havelẳ0, 1 Forlẳ0,mlẳ0 and forlẳ1,mlẳ ỵ1, 0,1 Thus we may have the following Y m l ðq;fịorbitals, whose values fromTables 1.1 and 1.2are:
Since Y 0 0 is independent ofqandf, it follows that the probability density distribution for thes-orbital is spherically symmetrical.
The Y 0 1 ;1 functions representing p orbitals resemble a "dumbbell" shape, consisting of two spheres centered on the z-axis and tangent at the origin There are three p orbitals, oriented at right angles to each other, which exhibit a distinct directional character denoted by the suffixes p x, p y, and p z Excluding the portion of the electron cloud near the origin that does not contribute to bond formation, one half of the dumbbell displays a positive wave function while the other half shows a negative wave function.
In the context of quantum mechanics, the magnetic quantum number \( m_l \) for d orbitals can take on five values: 0, 1, 2, leading to the formation of five distinct d orbitals Among these, four orbitals are characterized by their similarity, each featuring four pear-shaped lobes that are tangent to two others, with all lobes lying in a single plane aligned between a pair of axes These planes correspond to the xy-, xz-, and yz-planes, while the fourth orbital is centered on the x and y axes The fifth d orbital is unique, consisting of three regions of high probability density, which includes a toroidal shape complemented by two symmetrically placed pear-shaped regions along the z-axis.
The shapes of atomic orbitals in one-electron atoms are linked to three-dimensional spherical harmonics, as shown in Tables 1.1 and 1.2 These orbitals are not unique; any linear combination of them is also valid For instance, it is possible to create sets of orbitals where all the d-orbitals share the same shape, similar to how the px, py, and pz orbitals maintain identical shapes.
Electron Spin
Spin Orbitals
In a complete quantum mechanical description of the motion of an electron, both its position and spin coordinate must be considered Thus, its wavefunction must be a function ofx;y;zorr;q;f, ands:
Taking cognizance of the independence of spatial and orbital motions, we can set up wavefunctions as products of the spatialJnlmðr;q;fịand spin functions um sðsị;
Vðr;q;f;sị ẳJnlmðr;q;fịum sðsị (1.10.8) Orbitals likeF(r,q,f,s) are known as spin orbitals. z-axis
FIGURE 1.6 Orientations of electron spin vector with respect to the z -axis.
Since the quantum numbermstakes only two values 1 2 and 1 2 , we may have two spin functionsu 1 2 ðsị and u 1 2 ðsị Pauli definesu 1 2 ðsịasa(s) andu 1 2 ðsịas b(s).
Thus, for the ground state of hydrogen, the spin orbitals are
V1sẳJ1sðrịaðsị or V 0 1s ẳJ1sðrịbðsị (1.10.9)
In the excited state of a two-electron atom, such as helium, one electron occupies the 1s state while the other is in the 2s state.
V1sẳJ1sðrịaðsị or V 0 1s ẳJ1sðrịbðsị and
V2sẳJ2sðrịaðsị or V 0 2s ẳJ2sðrịbðsị (1.10.10) From these spin orbitals, we can obtain four possible wavefunctions for this partic- ular configuration of the helium atom.
V1ẳV1sV2sẳJ1sðr1ịaðs1ịJ2sðr2ịaðs2ị (1.10.11a)
V2ẳV1sV 0 2s ẳJ1sðr 1 ịaðs 1 ịJ2sðr 2 ịbðs 2 ị (1.10.11b)
V3ẳV 0 1s V2sẳJ1sðr1ịbðs1ịJ2sðr2ịaðs2ị (1.10.11c)
V4ẳV 0 1s V 0 2s ẳJ1sðr1ịbðs1ịJ2sðr2ịbðs2ị (1.10.11d) Since the two electrons are indistinguishable, a similar set of equations can be written by considering electron 2 in 1sorbital and electron 1 in 2sorbital.
Linear Vector Space and Matrix Representation
Dirac’s Ket and Bra Notations
In quantum mechanics, each dynamical state is characterized by a specific vector known as a ket vector, denoted as |ψ⟩ To differentiate between various kets, a label is added, such as |A⟩.
Thus, we define a ket as jJi ẳ
Ket’s form a linear vector space Any linear combination of several kets is also a ket vector. c1jAi ỵc2jBi ẳ jDi
In a dynamical system, each state uniquely corresponds to a specific direction of ket vectors, meaning that the states represented by jAi and cjAi, where c is a complex number, indicate the same physical state.
In linear algebra, each vector space has a corresponding dual vector space, allowing for the calculation of a scalar product between vectors from both spaces The vectors from the dual space of ket vectors are referred to as bra vectors, denoted as ⟨j| Therefore, the relationship can be expressed as ⟨j| = c₁, c₂, , cₙ.
The scalar product between the bra vector \( hB | \) and the ket vector \( | A \rangle \) is represented as \( hB | A \rangle \), resulting in a complex number In a dynamical system, the state at any given moment can be defined by the orientation of either a bra vector or a ket vector This relationship is expressed as \( | A \rangle + | B \rangle / hA | + hB | \) and \( c | A \rangle / c hA | \) Additionally, it is noted that \( hA | B \rangle = hB | A \rangle \).
Sometimes, in superficial treatments of Dirac notation, the symbol hJajJbi is alternatively defined as hJajJbi ẳ
Often only the subscript of the vector is used to denote a bra or ket ThusEq (1.11.12) may also be written as hajbi ẳ
The overlap integral between two functionsJaandJb therefore, be written as
Operation of an operator on a ket produces another ket vector Thus
In quantum mechanics, when performing operations on kets, the operator is positioned to the left of the ket Conversely, when operating on bras from the right with an operator A, the result is another bra The Schrödinger equation, represented as \( H | \Psi \rangle = E_n | \Psi \rangle \) in this notation, illustrates this relationship.
HbjJni ẳE n jJni or bHE n jJni ẳ0 (1.11.13) and the matrix elementsHnmof Hb are
(1.11.14) or in short notation, it is also written as
Similarly, the expectation value of an operatorAin the eigenstateJnmay be written as hAi ẳ
Atomic Units
Atomic units (au or a.u.) are employed in quantum mechanics to streamline equation representation and minimize the use of powers of ten There are two primary types of atomic units: Hartree atomic units and Rydberg atomic units, which differ based on the selection of mass and charge units This article focuses on Hartree atomic units, which are particularly advantageous for quantum mechanical calculations due to their numerical values aligning with four fundamental physical constants: electronic mass (m_e), electronic charge (e), and reduced Planck’s constant (ħ = h/2π).
Keẳ 4 pε 1 0 are taken as unity by definition.
In Hartree unity, atomic units define the unit of length as the Bohr radius (a₀), corresponding to the radius of the first Bohr orbit of the hydrogen atom, while the unit of energy is the Hartree (Eₕ), equivalent to twice the energy of hydrogen's ground state It is customary to express fundamental quantum chemistry equations in dimensionless form, with reduced length (r₀) defined as a/r₀ and reduced energy (E₀) as E/Eₕ When adopting atomic units, constants such as h, mₑ, e, and 4πε₀ take on the numerical value of 1, simplifying the Schrödinger equation for hydrogen-like atoms.
Jðrị ẳEJðrị (1.12.1) can be rewritten in a simplified form as
The atomic units of some common physical quantities and their equivalents in cgs and SI units are given inTable 1.4.
Approximate Methods of Solution of Schrửdinger Equation
Perturbation Theory
Perturbation theory is a valuable method applied when the wave equation diverges slightly from the true equation, typically by neglecting minor terms that have a minimal impact on the system This approach effectively addresses various complex problems, such as the hydrogen atom in electric or magnetic fields, the helium atom, the anharmonic oscillator, and Møller–Plesset corrections in molecular orbital theory.
Perturbation theory involves dividing the Hamiltonian into two components: one that can be solved and another that cannot This approach is effective only when the perturbation effects are minimal, allowing for accurate approximations in complex systems.
We assume that the HamiltonianHb can be expanded in terms of some parameterl, yielding
Hb ẳHb 0 ỵlHb ð1ị ỵl 2 Hb ð2ị ỵ.; (1.13.1) wherelcan be so chosen that the equation to which the above equation reduces when l/0 can be solved directly.
As l/0, the equation reduces to Hb ẳHb 0 and hence the Schroădinger equation becomes
This equation is said to be the wave equation for the unperturbed system, while the termslHb ð1ị ỵl 2 Hb ð2ị are called the perturbations.
The problem is solved in two steps In the first step, the eigenfunctions and eigen- values of the unperturbed Hamiltonian Hb ð0ị are obtained: b ð0ị
Table 1.4 Atomic units and equivalents in cgs and SI units
Quantity Unit cgs Equivalent SI Name
Angular Momentum _ h ẳ 1 1.05 10 27 erg s 1.05457 10 34 Js “ h-bar ”
Length a 0 ẳ _ 2 h m e e 2 ẳ 1 5.29 10 9 cm 5.2918 10 11 m Bohr or “ atomic unit ”
_ h ẳ 1 2.188 10 8 cm/s 2.1877 10 6 m/s Velocity in first Bohr orbit
_ h 2 ẳ 1 4.36 10 11 ergs or 627.509 kcal/mol or 27.211 eV
Electric fi eld e a 2 0 ẳ 1 5.142 10 9 V/cm 5.142 10 11 V/m Internal fi eld of H atom Electric constant 1 K e ẳ 1
In the second step of perturbation theory, eigenfunctions and eigenvalues are adjusted to incorporate the effects of perturbations These corrections are represented as an infinite series of terms that diminish progressively for well-behaved systems.
JnẳJ ð0ị n ỵlJ ð1ị n ỵl 2 J ð2ị n ỵ (1.13.5) Quite frequently, the corrections are only taken through first or second order (i.e., superscripts (1) or (2)) According to perturbation theory, the first-order correction to the energy is
J ð0ị n Hb ð1ị J ð0ị n ds (1.13.6) and the second-order correction is
Here, J ð0ị n is the normalized zeroth-order wavefunction and J ð1ị n is the first-order correction to the wavefunction.
In order to calculate the second-order correction in energy we need to knowJ ð1ị n Which can be written in terms of the zeroth-order wavefunction as:
Substituting this in the expression forEn ð2ị (Eq (1.13.7)), we obtain
As, an example, we can consider the case of an anharmonic oscillator with HamiltonianHb given by the equation
Comparison with Eq (1.13.1) shows that the unperturbed Hamiltonian Hb ð0ị is the same as that for a harmonic oscillator
The perturbation described by Hb ð1ị ẳax 3 ỵbx 4 (1.13.12) indicates that when the constants a and b are small, the eigenfunctions and eigenvalues of the anharmonic oscillator will closely resemble those of the harmonic oscillator.
The exact solution of the harmonic oscillator problem gives for the ground state energy and wavefunction the expressions:
The first-order correction to the ground state energy shall therefore be
In the integral of the equation, the first term disappears due to the odd nature of the integrand, resulting in no contribution from the ax^3 term to the harmonic oscillator's energy Conversely, the second term, which includes bx^4, contributes a value of 4a^3b^2, affecting the ground state energy of the oscillator Therefore, the total corrected ground state energy of the harmonic oscillator, representing the energy of the anharmonic oscillator, is derived from this contribution.
Variation Method
The variation method is a more robust approach compared to the perturbation method, particularly in situations where identifying an appropriate unperturbed Hamiltonian is challenging This method relies on a fundamental theorem that enhances its applicability in complex scenarios.
If J is any well-behaved wavefunction such that R
J Jdsẳ1, and if the lowest eigenvalue of the operator Hb isE 0 , then,
The choice of the variation function J is crucial; selecting it wisely allows the energy E to closely approximate the actual energy E0 If we use the true function J0 corresponding to the lowest state, the energy E will equal E0.
The variation method requires a trial wavefunction that includes adjustable parameters known as "variational parameters." These parameters are fine-tuned to minimize the energy of the trial wavefunction The optimized trial wavefunction and its associated energy provide upper bound approximations for the exact wavefunction and energy.
In practice, a trial wavefunction Jis expanded as a linear combination of a set of exact functionsFi, which may preferably form an orthonormal set.
JẳX N iẳ1 ciFi (1.13.18) wherecirepresents the set of variational parameters.
If the wave equation for the system under consideration isHbJẳEJ, we get
The equation can be simplified by using notation
In order to find values of the variation parametersc 1 ,c 2 ,.cNthat minimize the energy we differentiateEq (1.13.22)with respect to the variational parameters and impose the condition vE vck ẳ0 for kẳ1;2;.N
This leads to set ofEq (1.13.23)that will provide nontrivial solution, if the determi- nant constructed from them equals 0.
H N1 ES N1 H N2 ES N2 H NN ES NN
If an orthonormal set of functionsfiis used such thatR
F iFjdsẳdij, then dijẳ1 for iẳjand 0 ifisj In this case, the secular determinant reduces to
The secular determinant for N-basis functions results in an N-order polynomial in E, which can be solved for various roots, each corresponding to an approximate eigenvalue Subsequent chapters will explore several applications of variation theory.
Molecular Symmetry
Symmetry Elements
A symmetry element refers to a point, line, or plane that serves as the basis for a symmetry operation There are five distinct types of symmetry elements, each corresponding to specific symmetry operations.
1 Identity (E) It is a trivial symmetry element, which corresponds to doing nothing.
It is introduced for the purposes of mathematical group theory and is possessed by all molecules.
2 p-field rotation axis of symmetry designated asCp It corresponds to rotation through an angle 360 p Taking the example of water (H2O), a rotation by 180 about an axis dividing the HOH angle transforms the molecule to itself So, we say it has a twofold axisC2 Similarly, benzene (C6H6) has one sixfold axisC6perpendicular to the molecular plane and six twofold axesC 2 in the molecular plane The axis having the largest value ofpis called the principal axis.
3 Plane of symmetry—usually designated ass with subscriptsv,h, orddepending on whether the plane is a vertical, horizontal, or diagonal plane of symmetry This corresponds to reflection in a mirror plane If the plane contains the principal axis, it is called vertical,sv If the plane is perpendicular to the principal axis, it is called horizontal,sh The diagonal plane,sd, is a vertical plane that bisects the angle between twoC 2 axes Thus, H 2 O has two vertical planes of symmetry designated as svands 0 v (Figure 1.7), and CH3Cl has three.
4 Center of symmetry—designated asi A molecule has a center of symmetry, i, if by reflection at the center, the molecule transforms into a configuration indistinguish- able from the original one This operation transforms all atoms with coordinates (x,y,z) to identical atoms with coordinate (x,y,z) Typical examples are benzene, carbon dioxide.
5 p-fold rotation–reflection axis of symmetry designated asS p This is a combination of two successive operations—a rotation through 360 p followed by reflection at a plane perpendicular to the axis of rotation Neither operation alone need to be a symmetry operation Thus, CH4molecule has threeS4axes, and borontrifluoride has oneS3axis.
Symmetry Point Groups
A point group is defined as a set of operations, including the identity operation (E), that leaves a molecule unchanged According to group theory, only specific combinations of symmetry elements that adhere to certain rules can exist These combinations must maintain at least one point unchanged Molecules such as H₂O, F₂CO, and CH₂Cl₂ are categorized under the same point group, C₂v, indicating they share identical symmetry descriptions Essential criteria must be met for symmetry elements to form a valid point group.
1 One of the operations in the group is identityE, for whichEAẳAEẳA, where A represents an element in the group.
FIGURE 1.7 Symmetry elements for water molecule —C 2 ( z ) rotational axis and s v and s 0 v planes of symmetry.
2 For every element in a group there is also an inverseA 1 such thatA 1 AẳI, where
Iis a unit matrix Thus, ifAbelongs to a group then, A 1 ẳB will also belong to the same group.
3 The symmetry operations are associative, that is
4 Successive application of two symmetry operations is also a symmetry operation.
A twofold rotation (C2) of a point's coordinates (x, y, z) around the z-axis, followed by a reflection across the xz-plane, yields a point that is equivalent to reflecting the original point across the yz-plane.
Thus,ðX;Y;Zị! C 2 ðZị ðX;Y;Zị! sðxzị ðX;Y;Zị
Also,ðX;Y;Zị sðyzị !ðX;Y;Zị
In general however, ABsBA as may be seen from the following symmetry transformations: ðx;y;zị! C 4 ðzị ðy;x;zị C 2 ! ðxị ð y;x;zị and ðx;y;zị! C 2 ðxị ðx;y;zị! C 4 ðzị ðy;x;zị
So, in this case,C2(x)C4(z)sC4(z)C2(x)
In the context of C2v symmetry, the impact of successive symmetry operations can be illustrated using a multiplication table This symmetry consists of four elements: E, C2(z), sv(xz), and s0v(yz) As demonstrated in Table 1.5, these elements satisfy the criteria necessary to form a group.
Table 1.5 can easily be verified for water Figure 1.7 It can be seen that the product of any two symmetry transformations leads to another member of the group.
Table 1.5 Multiplication table forC 2v symmetry operations a
C 2 (z) C 2 E s 0 v s v s v (xz) s v s 0 v E C 2 s 0 v ðyzị s 0 v s v C 2 E a The fi rst operation at the top row to be followed by the second operation at the left column.
Classification of Point Groups
We can now list the possible symmetry point groups of molecules:
1 Groups with noCpaxis In this case the molecule has no symmetry element except
E The molecules are then said to belong to group C1 But, if has a plane of sym- metry it belongs toCsgroup.
2 Groups with a singleCpaxis Molecule with a singleCpaxis belongs to groupCp
(pẳ2, 3, 4.) Thus, H2O 2 and CH 2 Cl–CH 2 Cl (gauche) belong toC 2 group.
Molecules that possess an identity element and a p-fold axis of rotation, along with p vertical planes of symmetry that include the rotation axis, are classified within the Cpv group For example, ammonia is categorized in the C3v group due to its symmetry elements, which include the identity element E.
C3and threesvplanes Some other molecules belonging to this group are CHCl3, CCl3–CBr3, etc.
Molecules that exhibit a horizontal plane of symmetry in addition to having E and Cp axes belong to the Cph symmetry group Examples of such molecules include H2O and trans-ClH2C–CH2Cl, which fall under the C2h category, while H3BO3 and B(OH)3 are classified within the C3h group.
A molecule with only an ap-fold rotation–reflection axis of symmetry is classified as belonging to the Sp group Additionally, molecules that possess a center of symmetry are referred to as belonging to the Ci group, such as trans-dichlorodibromoethane.
Linear molecules lacking a plane of symmetry perpendicular to their molecular axis are classified under the point group Cₙ, as they exhibit an infinite-fold axis and numerous planes of symmetry Notable examples of this group include OCS, OCN, and HCN Conversely, linear molecules that possess a center of symmetry fall into the Dₙh point group, such as dicyanodiacetylene.
3 Groups withp-fold principal axisCpandpC2axes perpendicular to theCpaxis constitute theDp,Dph, andDnd group ForDpgroup, the symmetry elements areCp axis andnC2axes perpendicular to theCpaxis.D1is of course equivalent to C2and the molecules of this symmetry are classified asC2.
If in addition to D p operations, the molecule also has a horizontal symmetry planesh, it belongs toDnhgroup Benzene (C6H6) has the elementsE,C6, 3C2, 3C 2 0 , andshand so belongs to D 6h group.
Molecules that exhibit Dp operations and possess diagonal mirror planes (sd) that intersect the p-fold axis while bisecting the angles between consecutive C2 axes are classified within the Dpd group An example of this is allene (H2C=C=CH2), where one CH2 group is rotated 90 degrees relative to the other, categorizing it in the D2d group due to its symmetry elements, which include E, three C2 axes, S4, and two sd planes.
4 Groups with more than oneCpaxis,p>2 Many important molecules such as methane, CCl 4 , SF 6 , etc., have more than one principal axes Thus, methane has four threefold axes (4C3) and three mutually perpendicular twofold axes (2C2) Such molecules form cubic groups such as tetrahedral group (T,Td,Th) and octahedral group (O,Oh) They possess rotational symmetry of tetrahedron or octahedron.
Molecules are classified into the T group if they possess four threefold axes and three mutually perpendicular twofold axes When these molecules also feature a center of symmetry, they are categorized as belonging to the Th group Furthermore, molecules that exhibit the symmetry elements of the T group, along with two mutually perpendicular planes of symmetry through each twofold axis—resulting in a total of six planes of symmetry—are classified under the Td group A prime example of this is methane (CH4), which has symmetry elements including 3C2, 4C3, and 3S4, along with six undesignated planes of symmetry.
Molecules in the O group exhibit three mutually perpendicular C4 axes and four C3 axes, whereas those in the Oh group also possess a center of symmetry and nine planes of symmetry Notable examples of the Oh symmetry group include SF6 and (COCl6)4.
Representation of Point Groups and Character Tables
A representation of a group consists of a set of matrices that correspond to individual symmetry operations, adhering to the group's multiplication rules These matrices can be one-, two-, or three-dimensional and are often simplified into a set of numbers known as traces, which are the sums of their diagonal elements While numerous representations exist, not all are mutually independent; the simplest set of independent representations is termed an irreducible representation The entries within a representation are referred to as characters, and the compilation of these irreducible representations forms a character table For instance, the character table of the C2v symmetry group is illustrated in Table 1.6.
The table presents symmetry operations, including E, C2, sv, and s0, along with the irreducible representations A1, A2, B1, and B2 The characters þ1 and 1 denote symmetric and antisymmetric transformations, respectively The group order is indicated as hẳ4 in the sixth column, while the simple functions of coordinates x, y, and z are associated with specific irreducible representations These coordinates are essential for understanding the characteristics of normal vibrational, translational, or rotational modes within the group.
Table 1.6 Character table for theC2v point group
The wavefunction B 2 1 1 1 1 y, T y, R x can be associated with irreducible representations of a symmetry group, referred to as symmetry types or species, which are frequently utilized in molecular spectroscopy Standard notations in character tables are employed to denote these irreducible representations and the corresponding symmetry operations.
A—for symmetric with respect to the principal axis of symmetry
B—for antisymmetric with respect to the principal axis of symmetry
E—for doubly degenerate vibrations or wavefunctions These are represented by
F—for triply degenerate vibrations or wavefunctions.
In order to differentiate among the various representations of the same type, some subscripts or superscripts are used These are:
Subscripts 1 and 2—to represent symmetric or antisymmetric with respect to the plane of symmetry.
Superscripts prime ( 0 ) and double prime ( 00 )—to represent symmetric or antisym- metric with respect to a plane of symmetry.
Subscriptsgandu—to represent symmetric or antisymmetric with respect to center of symmetry.
In case of linear molecules having symmetry group C N v or D N h, the notations used are:
S þ —to represent symmetric with respect to plane of symmetry through the molec- ular axis
S —to represent antisymmetric with respect to plane of symmetry through the molecular axis p,D,f—to represent degenerate vibrations or wavefunctions of order (degree 2, 3, or 4), in increasing order.
In addition, subscripts g and u are used to show symmetry with respect to center of inversion.
In this article, we will explore the concepts of representations and characters for the C2v group, illustrated through examples of normal vibrational modes and the electronic wavefunctions of the water molecule.
1.14.4.1 Symmetry of Normal Vibrations of Water Molecule
Water is a triatomic molecule classified under the C2v point group, exhibiting three degrees of vibrational, rotational, and translational freedom Focusing solely on its vibrational modes, we can analyze the impact of symmetry operations such as E, C2(z), sv(xz), and sv(yz), with the y-z plane designated as the molecular plane.
The analysis indicates that symmetry operations do not influence the shapes of Q1 and Q2; however, for Q3, the symmetry operations C2 and sv invert the displacement vectors.
In the context of symmetry operations, transformation numbers indicate the effect on nondegenerate vibrations, where normal coordinates can be either symmetrical (+1) or antisymmetrical (-1) for each vibrational mode The normal modes Q1 and Q2 share the same representation G1, while mode Q3 is represented by a different representation, G2.
A comparison with Table 1.6 shows that G1 and G2 belong to species A 1 and B 2 , respectively Thus, water molecule has three vibrational modes belonging to species 2A 1 þB 2
FIGURE 1.8 Symmetry operations on the Q 1 , Q 2 , and Q 3 vibrational modes of water Arrows show displacement vectors for the three atoms.
1.14.4.2 Symmetry of Electronic Orbitals of Water Molecule
This article examines the impact of the C2v group operations on the 2s, 2px, 2py, and 2pz orbitals of the oxygen atom, as well as the 1s orbitals of the hydrogen atoms in a water molecule The influence of the C2(z) operation on these orbitals is illustrated in Figure 1.9.
The effect of the symmetry operationsE;C2ðzị;svðxzị; ands 0 vðyzịof theC2vgroup is as follows:
Eð2sị/2s; C 2 ð2sị/2s; svð2sị/2s; s 0 vð2sị/2s Eð2pxị/2px; C2ð2pxị/2px; svð2pxị ẳ2px; s 0 vð2pxị ẳ 2px
Eð2pyị/2p y ; C 2 ð2pyị/2p y ; svð2pyị/2p y ; s 0 v ð2pyị/2p y
Eð2p z ị/2pz; C 2 ð2p z ị/2pz; svð2p z ị/2pz; s 0 vð2p z ị/2pz
The transformation numbers for these operations can be represented in the following tabular form:
FIGURE 1.9 Symmetry transformation of 2s and 2p orbitals of oxygen and 1s orbital of hydrogen under C 2 (z) operation (2 p x not shown).
The numbers in each row correspond to the group multiplication table of the C2v group, indicating that each represents an irreducible representation denoted by the symbol G Notably, since the representations G1 and G4 are identical, the C2v group ultimately yields only three distinct representations.
A comparison of the operations with the character table for C2v groups reveals that the orbitals O2s and O2p z correspond to the irreducible representation A1, while O2px is associated with the B1 representation Consequently, any molecular orbital formed from the atomic orbital O2px will be classified as a b1 orbital In a similar manner, O2py is linked to the B2 representation and can contribute to a b2 molecular orbital The letters a1, a2, b1, and b2 are utilized to denote the symmetry of atomic orbitals in polyatomic molecules belonging to the A1 representation.
In the case of two hydrogen atoms, H1 and H2, represented by 1s orbitals, symmetry operations can be expressed through matrices that follow the multiplication rules of the C2v group Notably, the identity operation, E, signifies that no changes occur within the system.
TheC 2 (z) rotation interchanges the positions of H 1 and H 2 , so
It can be seen that the (22) matrices representing the operationsE,C2,sv, ands 0 v satisfy the multiplication rule of theC2vgroup and so constitute a representation.
Symmetry Properties of Eigenfunctions of Hamiltonian
The mathematical relationship between the eigenfunctions of the Hamiltonian and group representations is rooted in the invariance of the molecular Hamiltonian under three distinct symmetry operations.
1 Symmetry arising out of permutation of electrons Due to indistinguishability of electrons their permutation changes identical terms in the Hamiltonian, leaving theHamiltonian unchanged.
2 Spin symmetry arising out of the fact that the Hamiltonian does not contain spin- dependent terms.
3 Spatial symmetry A symmetry operation such as rotation about an axis of symme- try or reflection about a plane of symmetry interchanges identical nuclei leaving the Hamiltonian invariant.
In the case of the symmetry operation R, such as the interchange of hydrogen atoms in H2O, the potential field affecting the electrons remains unchanged Consequently, the Hamiltonian H of H2O remains invariant under this symmetry operation.
So, the function RJk is also an eigenfunction ofHwith the same eigenvalue Ek It, therefore, follows that ifEkis a nondegenerate eigenvalue with eigenfunctionsJkthen
RJkmust either be equal toJkor just differ from it by a change of sign, i.e., RJkẳJk orJk IfEkis anm-fold degenerate eigenvalue with orthonormal set of eigenfunctions
J ðiị k ðiẳ1;2;.mị then RJk will also belong to the same set or shall be some linear combination of the members of set.
C ki J ðiị k (1.14.4) whereJ ð1ị k ;J ð2ị k ;.J ðmị k are members of the degenerate setJk Same conclusions can be derived by using the properties of commuting operators (Section 1.6.2).
The eigenfunctions are therefore restricted by the symmetry of the molecule It can be proved that they have the symmetry properties of the irreducible representations of the group.
The unique characteristics of eigenfunctions play a crucial role in the construction of molecular orbitals and in predicting both allowed and forbidden transitions Additionally, group theory serves as a significant mathematical framework, contributing to various essential applications in the fields of chemistry and spectroscopy.
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2.1 Wavefunction of Many-Electron Atoms 47 2.2 Slater Determinants for Wavefunctions 50 2.3 Central Field Approximation 51 2.4 Self-consistent Field (SCF) Approximation—Hartree Theory 52 2.4.1 Hartree–Fock Method 53 2.4.1.1 Generalization of the HF method to a many-electron atom 55 2.4.2 Interpretation of the Eigenvalues of the Fock Operator 58 2.5 Electronic Configuration and Electronic States 58 2.6 Restricted and Unrestricted Wavefunctions 61 References 62 Further Reading 62
In Chapter 1 we discussed the development and solution of the time-independent Schro¨dinger equation for a one-electron atom We shall now extend the treatment to a many-electron system.
2.1 Wavefunction of Many-Electron Atoms
In a many-electron system, it is essential to incorporate electron repulsion into the potential energy term of the wave equation Consequently, the potential of a many-electron atom with a nuclear charge of +Ze is represented in atomic units.
In this system, the distance of the i-th electron from the nucleus is represented as \( r_i \), while \( r_{ij} \) denotes the interelectronic distance To prevent double counting of each \( r_{ij} \) term in the summation, a factor of \( \frac{1}{2} \) is incorporated in the second term The Hamiltonian, denoted as \( H \), characterizes the overall energy of the system.
The equation includes three key components: the first term represents the kinetic energy operator of the electron, the second term accounts for the attraction between the electron and the nucleus, and the third term signifies the cumulative effect of repulsions between two electrons.
The Schro¨dinger equation for anN-electron atom may then be written as
The solution to the equation presented in Eq (2.1.2) is complex due to electron repulsion terms, making it difficult to separate variables as in the case of a hydrogen atom However, by assuming that the electrons operate independently, we can simplify the Hamiltonian, H, to the sum of individual one-electron Hamiltonians.
To approximate a solution for Eq (2.1.3), we will disregard the electron repulsion term This allows us to express the equation as a sum of individual one-electron Hamiltonians, represented as H1, H2, , HN Consequently, the equation can be rewritten as (H1 + H2 + + HN)ψ(1, 2, , N) = Eψ(1, 2, , N).
We can now solve the equation by the standard “separation of variables” technique used earlier for the hydrogen atom problem.
Jð1;2; Nị ẳF1ð1ịF2ð2ị.FNðNị (2.1.6) whereF1(1),F2(2),.F N (N) are the one-electron orbitals and F i (j) shows that j th elec- tron is in thei th orbital SubstitutingEq (2.1.6)into(2.1.5), we get
F N ðNịHNFNðNị ẳE (2.1.7) Each term on the left-hand side is independently variable and so each of them must individually be equal to a constant.
HNF N ðNị ẳENF N ðNị Thus,E ẳE1ỵE2ỵ.EN and,
HJð1;2; Nị ẳ ðE1ỵE2ỵ.ENịJð1;2; Nị (2.1.8) Here,Jð1;2;.Nịis defined byEq (2.1.6).
In constructing a many-electron wavefunction, we can represent it as a product of orbitals, known as the Hartree product However, the presence of electron repulsion complicates this, as the many-electron Hamiltonian cannot be expressed merely as a sum of one-electron operators The interaction term, which relies on the instantaneous relative positions of electrons, prevents the total wavefunction from being a straightforward product of orbitals Therefore, it is essential to incorporate electron spin, necessitating the use of spin orbitals in the formulation of the many-electron wavefunction.
When analyzing electronic wavefunctions, it is crucial to consider the symmetry property related to the interchange of electron coordinates, commonly known as Pauli’s exclusion principle Electrons are indistinguishable particles, meaning that the physical properties of a system remain unchanged regardless of how we renumber or rename the electrons Consequently, the many-electron density function, denoted as r(1, 2, , N), is invariant under the exchange of any two electrons, as expressed by the relation r(1, 2, , N) = |Ψ(1, 2, , N)|².
The indistinguishability of electrons results in specific symmetry properties of the wavefunction When applying a permutation operator Pij that swaps all coordinates, including spin, of electrons i and j, the wavefunction exhibits these symmetry characteristics.
P ij 2 Jð1;2; i;j Nị ẳPijJð1;2; j;i Nị ẳJð1;2; i;j Nị (2.1.10) orPijẳ 1
When two electrons are interchanged, the wavefunction, J, can change by a factor of +1 (symmetric) or -1 (antisymmetric) These two outcomes are the only options that maintain the invariance of J² The antisymmetric nature of the wavefunction is particularly relevant for electrons, as it gives rise to Pauli's exclusion principle, which asserts that no two electrons can occupy the same spin orbital.
A single product function does not meet the antisymmetry principle, making it unsuitable for approximation However, a combination of two functions can fulfill this requirement For instance, in a two-electron atom like the excited helium atom, where the lowest energy orbitals are 1s and 2s, either electron can occupy the 1s or 2s orbital due to their indistinguishable nature.
If there is only one electron in each orbital, then on ignoring electron spin, the two- electron wavefunction is
Jð1;2ị ẳF1sð1ịF2sð2ị
If we apply the two-electron permutation operator, then
P12Jð1;2ị ẳF1sð2ịF2sð1ị
Clearly,P12J(1,2) is not the negative of J(1,2) However, a combination of Hartree products may be constructed which has antisymmetry Consider now a wavefunction like
Jð1;2ị ẳF1sð1ịF2sð2ị F1sð2ịF2sð1ị (2.1.12)Operation by the operatorP12shall, therefore, give
P12Jð1;2ị ẳF1sð2ịF2sð1ị F1sð1ịF2sð2ị ẳ ẵF1sð1ịF2sð2ị F1sð2ịF2sð1ị ẳ Jð1;2ị (2.1.13)
The wavefunction, therefore, changes sign The correct form of wavefunction for the two-electron system under consideration will therefore be given byEq (2.1.12).
When incorporating electron spin into our analysis, we can denote the electron in the 1s orbital with spin up as F1s(1)a(1) and the electron in the 2s orbital with spin down as F2s(2)b(2) Consequently, the Hartree product for a singlet state can be expressed accordingly.
Jð1;2ị ẳF1sð1ịað1ịF2sð2ịbð2ị (2.1.14a) and
Jð2;1ị ẳF1sð2ịað2ịF2sð1ịbð1ị (2.1.14b)
We may then construct an antisymmetric function as
JẳF1sð1ịað1ịF2sð2ịbð2ị F1sð2ịað2ịF2sð1ịbð1ị (2.1.15a) or in the normalized form
Jẳ 1 ffiffiffi2 p ẵF 1s ð1ịað1ịF2sð2ịbð2ị F1sð2ịað2ịF2sð1ịbð1ị (2.1.15b)
This will be the correct form of antisymmetric wavefunction inclusive of electron spin The wavefunction can also written in matrix form as