Quantum Chemistry PDF

# QUANTUM CHEMISTRY ## Introduction: - De Broglie Hypothesis, Heisenberg's uncertainty principle, quantization of energy, Operators, Schrodinger wave equation, well-behaved function - Particle in a one-, two and three-dimensional box - Physical interpretation of the wave function and probability d...

# QUANTUM CHEMISTRY ## Introduction: - De Broglie Hypothesis, Heisenberg's uncertainty principle, quantization of energy, Operators, Schrodinger wave equation, well-behaved function - Particle in a one-, two and three-dimensional box - Physical interpretation of the wave function and probability densities for ID box, degeneracy - Applications to conjugated systems, zero-point energy and quantum tunnelling - Numerical calculations ## Introduction: Quantum mechanics (or wave mechanics or particle mechanics) is a branch of science which takes into consideration the de Broglie's concept of dual nature of matter and Planck's quantum theory. It explains the phenomenon related to small particles. ## The Wave Mechanics: - The wave mechanics was put forward by Schrodinger in 1926. It is based on dual nature of matter and is applicable only for small microscopic particles. - Quantum mechanics deals with microscopic particles only and not the macroscopic ones. - Takes into consideration de Broglie hypothesis and Heisenberg’s uncertainty principle, which is dependent on Planck’s quantum theory according to which only discrete values of energy are absorbed or emitted. - Schrodinger wave equation is considered the heart of quantum mechanics. ## Quantum Mechanics and Probabilities: - Quantum mechanics provides information about the probabilities of finding microscopic particles (such as electrons) at various locations within the space. - This makes studying quantum mechanics essential. ## De Broglie's hypothesis: - According to de Broglie, small particles in motion act as wave and particle (i.e. they exhibit dual nature). - The energy associated with a photon of frequency *v* is given by the relation *E = hv*. - By using the Einstein’s mass energy relationship *E = mc²*, and equating it with *E = hv* we get *mc² = hv = hc/λ*. - The equation *mc = h/λ* can also be written as *mass x velocity = h/λ* which translates to *momentum (p) = h/λ* or *p ∝ 1/λ*. - This final equation (p ∝ 1/λ) is known as the de Broglie equation – momentum of a particle on motion is inversely proportional to wavelength. ## Heisenberg's uncertainty principle: - According to Heisenberg, it is impossible to determine accurately or precisely both the position and momentum of a moving microscopic particle simultaneously. - This is known as Heisenberg’s uncertainty principle. - Mathematically, it is expressed as: *Δp × Δx = h/ 4π* or *m × Δv = h/ 4π* - Certainty in the measurement of position introduces uncertainty in the measurement of momentum and vice versa. These properties are called conjugate properties. - The Heisenberg’s principle is only important in considering measurements of small particles comprising of atomic particles and is negligible in case of large or macroscopic objects. ## Bohr's atomic model: 1. Electrons revolve around the nucleus only in specific permitted circular orbits. The orbits with *n = 1, 2, 3,...* are designated as K, L, M, N etc orbits. 2. When present in these orbits, the electrons do not radiate energy. 3. Electrons can move from one orbit to the other by quantum or photon jumps only. - The change in energy ΔE between the higher energy state and lower energy state is calculated by the formula: ΔE = E_high - E_low = hv. - *h* is Planck's constant and *v* is the frequency of the emitted radiation. - Angular momentum of an electron (mvr) orbiting around the nucleus is an integral multiple of *h/2π*. This is expressed as *mvr = n(h/2π)*. - *m* is the mass of electron, *v* is its velocity, *r* is the radius of the orbit, and *n* is the principal quantum number, which is an integer (n = 1, 2, 3,...). Therefore, the angular momentum of the orbiting electron is said to be quantized. ## Calculation of radius of orbits: - Consider an electron of charge (*e*) and mass (*m*) moving in a circular orbit of radius *r* with a velocity (*v*). - The electrostatic force of attraction between nucleus and electron is given by *F = Ze²/r²*. - The centrifugal force acting on the moving electron is given by *Centrifugal force = mv²/r*. - To keep the electron moving in the circular orbit, the electrostatic force between the electron and the nucleus and the centrifugal force must be equal. This can be expressed as *Ze²/r² = mv²/r*. - For hydrogen, *Z = 1*. Therefore, we can rewrite the equation as *e²/r² = mv²/r*. - From one of the postulates of Bohr's theory, *mvr = nh/ 2π*. This can also be rewritten as *v = nh/2πmr*. - By substituting *v* in the equation *e²/r² = mv²/r*, we get the following equation: *e²/r = m(nh/2π mr)²*. - Rearranging the equation, we get: *r = n²h²/4π²me²*. - This equation relates the radius of the orbit to the principal quantum number, Planck's constant, the mass of the electron, and the electronic charge. - Substituting values in the equation *r = n²h²/4π²me²*, we get *r = 0.529 n² × 10⁻⁸ cm*. - For *n = 1*, the value of *r* is *r = 0.529 × 10⁻⁸ cm*. This is called the Bohr’s radius. ## Energy of electron in the orbit: - The total energy of the moving electron is the sum of its kinetic energy and potential energy: *E = K. E. + P. E. = (1/2) mv² + (-e²/r)*. - The kinetic energy can be expressed as *mv² = e²/r*. - Therefore, total energy can be expressed as *E = ½.(e²/r) – e²/r = - (1/2) e²/r*. - Using the equation *r = n²h²/4π²me²*, we can further simplify the equation to *E = -2²*m*e⁴/ n²h²*. - This is the equation of energy of electron in the Bohr’s orbit. ## Schrodinger wave equation: - The Schrodinger wave equation is the keynote of wave mechanics and it is based on the concept of considering electron as a standing wave around the nucleus. - The equation of standing wave is: *Ψ = Asin(2πx/λ)*. - *Ψ* is the amplitude of the wave, *x* is the displacement in a given direction, *λ* is the wavelength, and *A* is a constant. - Differentiating the equation twice with respect to *x*, we get: *dΨ/dx = A(2π/λ)cos(2πx/λ)* and *d²Ψ/dx² = -A(4π²/λ²)sin(2πx/λ)*. - Using the equation *Ψ = Asin(2πx/λ)*, this final equation can be written as: *d²Ψ/dx² = -4π²/λ²Ψ*. - By the definition of kinetic energy (K.E.), It can be rewritten as: *K.E. = mv²/2 = (m²/2m)(v²)*. - Using the de Broglie’s equation *λ = h/mv*, we get *λ² = h²/m²v²* or *m²v² = h²/λ²*. - Substituting the value *m²v² = h²/λ²* in the equation *K.E. = (m²/2m)(v²)*, we get *K.E. = (1/2)(h²/λ²)*. - Using the equation *d²Ψ/dx² = -4π²/λ²Ψ*, we can rewrite the kinetic energy expression as: *K.E. = (h²/2m)(d²Ψ/dx²)*(1/4π²)(d²Ψ/dx²)*. - The total energy *K.E.* can also be represented as the sum of total energy (*E*) minus potential energy (*P.E*). This can be written as: *K.E = E - P.E*. - Rewriting the equation, we get *E - P.E. = (h²/8m)(d²Ψ/dx²)*. - We can rewrite this equation as: *d²Ψ/dx² = (8π²m/h²)(E - P.E)Ψ*. - This final equation is the Schrödinger wave equation in one dimension. - The equation can be generalized for a particle whose motion is described in three dimensions with coordinates *x, y, z* as: *∂²Ψ/∂x² + ∂²Ψ/∂y² + ∂²Ψ/∂z² = (8π²m/h²)(E - P.E)Ψ* - The equation can also be written in the following format: * (d²/dx² + d²/dy² + d²/dz²)Ψ + (8π²m/h²)(E - P.E)Ψ = 0* or *∇²Ψ + (8π²/h²)(E - P.E)Ψ = 0*. - ∇² = (∂²/∂x² + ∂²/∂y² + ∂²/∂z²) is called Laplacian operator. ## Eigen values and Eigen functions: - The Schrodinger wave equation is a second-degree differential equation, meaning there are several solutions. Some of these solutions are imaginary and not valid. - If the potential energy is known, the total energy and the corresponding values of wave function *Ψ* can be obtained. - The wave function *Ψ* is a finite, single valued and continuous function that is zero at infinite distance. - Eigen values refer to the values of total energy (which is a characteristic of a wave function) and meet the requirements of the wave function. - Corresponding to the Eigen values of total energy, the characteristic values of wave functions are called Eigen functions. - Eigen values correspond very nearly to the energy values associated with different Bohr’s orbits. Therefore Bohr’s atomic model can be considered as a direct consequence of the wave mechanical approach. - If an operator *A* operating on a function *Ψ*, converts it into the same function multiplied by a constant factor *a*, then the function *Ψ* is called an Eigen function and the corresponding values of the constant factor *a* are called Eigen values. - This equation is known as Eigen equation. - *A Ψi = α. Ψi* - The Schrodinger wave equation can be expressed in Eigen form as *Ηψ = Εψ*, where *H* is called the Hamiltonian operator. ## Significance of wave function: 1. In Schrodinger wave equation, *ψ* represents the amplitude of the spherical wave around the nucleus. 2. The probability of finding an electron is directly proportional to the square of the wave function, *ψ²*. 3. If the wave function is imaginary, *ψψ** becomes a real quantity, where *ψ*** is the complex conjugate of *ψ*. 4. For a hydrogen atom, Schrodinger wave equation gives the wave function of the electron situated at a distance *r*: *Ψ = C₁e⁻C₂r*, where C₁ and C₂ are constants. 5. When the electron gets excited, it is raised from *n* to higher energy levels (*n = 2, 3,...*), the solution of the wave equation gives a set of values *ψ²* which correspond to different shapes for the distribution of the electron (i.e. the concept of atomic orbitals). ## Hamiltonian operator (H): - The Hamiltonian operator (*H*) is defined as: *H^ = (h²/2m)∇² + V*, where *V* is the potential energy. - By using the Hamiltonian operator, the Schrodinger wave equation can be written as: *(h²/2m)∇² + V*Ψ = *EΨ* or *H^Ψ = EΨ*. - This is the simplified form of the Schrodinger wave equation. ## Particle in one dimensional box: - The particle in a one-dimensional box is the simplest quantum mechanical problem. - Consider a particle of mass *m* confined in a one-dimensional box of length *a* having infinitely high walls. - The potential energy (*V*) of the particle is assumed as zero inside the box: *V(x) = 0*. - The Schrodinger wave equation inside the box can be written as: - *(h²/2m)(d²/dx²) + V(x) ψ(x) = Eψ(x)* - *(h²/2m)(d²/dx²) = Eψ(x)* - *(d²/dx²) = (2m/h²) Eψ(x)* - *(d²/dx²) + (2m/h²) Eψ(x) = 0* - *(d²/dx²) + k²ψ(x) = 0* - The constant *k* can be calculated by: *k² = (2mE/h²)*. - The solution of this equation is of the form: *ψ = B sin kx*. - Using the boundary conditions (*ψ = 0* at *x = 0* and *x = a*), we get *sin ka = 0* or *ka = nπ*. - Therefore, *k = nπ/a*, where *n = 0, 1, 2, 3,...* is the quantum number. - The final equation for this problem is *ψ = B sin (nπx/a)*. - The accepted values for *n* are 1, 2, 3, … because *n = 0* means the particle is not inside the box. - From equations *(d²/dx²) + (2m/h²) Eψ(x) = 0* and *k = (2mE/h²)*, we get *E = E_n = (n²h²/8ma²)*. - This equation gives the energy of the particle in one dimensional box. - The energy depends on the quantum number (*n*), which has only integral values, therefore, there are only certain values of *E* (and of the electron), meaning it is quantized. ## Normalization of wave function (ψ) - The total probability of finding a particle within the box is 1. - The normalization of *ψ* requires that: *∫₀ᵃ ψ²dx = 1* - From the equation *ψ = B sin (nπx/a)*, we get *∫₀ᵃ B²sin²(nπx/a)dx = 1* - The integral of *∫₀ᵃ sin²(nπx/a)dx* is *(1/2)a*. - Using the normalization rule, we get *B²*(1/2)a =1* or *B = (√2/a)*. - The normalized wave functions for the particle in one-dimensional box are: *ψ(x) = (√2/a)sin(nπx/a)*. ## Particle in a 3-dimensional box: - Consider the motion of a particle of mass *m* confined to a three-dimensional box with edges *a, b*, and *c*. - The potential energy within the box is zero, and is infinite outside the box. - The time-independent Schrodinger equation for the particle is: - *(h²/2π²)(∂²/∂x² + ∂²/∂y² + ∂²/∂z²) ψ(x, y, z) = Eψ(x,y,z)* - *(∂²/∂x² + ∂²/∂y² + ∂²/∂z²)ψ = (2m/h²)Eψ* - *(∂²/∂x² + ∂²/∂y² + ∂²/∂z²)ψ + (2m/h²)Eψ = 0*. - The wave function *ψ(x, y, z)* is a product of three functions, each depending on just one coordinate: *ψ (x,y,z) = X(x)Y(y)Z(z)*. - Using this, we can rewrite the Schrodinger equation as: *YZ(∂²X/∂x²) + XZ(∂²Y/∂y²) + XY(∂²Z/∂z²) + (2mE/h²)ψ = 0*. - Dividing the equation by *XYZ*, we get *(h²/2m)(1/X)(∂²X/∂x²) + (1/Y)(∂²Y/∂y²) + (1/Z)(∂²Z/∂z²) + E=0*. - *E* is written as the sum of three contributions associated with the three coordinates, which can then be separated into the following three equations: - *(h²/2m)(1/X)(∂²X/∂x²) = Ex* - *(h²/2m)(1/Y)(∂²Y/∂y²) = Ey* - *(h²/2m)(1/Z)(∂²Z/∂z²) = Ez* - The solutions for these equations are similar to the equations of particle in one dimensional box. - *X(x) = (√2/a)sin(n_xx/a)* - *Y(y) = (√2/b)sin(n_yy/b)* - *Z(z) = (√2/c)sin(n_zz/c)* - The energy can then be expressed as: - *Ex = (n_x²h²/8ma²)* - *Ey = (n_y²h²/8mb²)* - *Ez = (n_z²h²/8mc²)*. - *n_x, n_y* and *n_z* are integers (≠0). - Therefore, the total energy of the particle is: *E = (h²/8m)(n_x²/a² + n_y²/b² + n_z²/c²)*. - If the box is cubical, the energy of the particle will be: *E = (h²/8ma²)(n_x² + n_y² + n_z²)*. - Degeneracy is a concept that arises from the quantum numbers. Because each state is characterized by three quantum numbers, it is possible for several excited states to have the same energy. This is called degeneracy. The levels are said to be degenerate when energy levels of different states are the same. - Degeneracy and energy levels are summarized in the following table: | Quantum Numbers and Number of States | Energy Level | Degeneracy | |---|---|---| | (111) | (3h²/8ma²) | Non-degenerate | | (211) (121) (112) | (6h²/8ma²) | Three-fold generate | | (221) (122) (212) | (9h²/8ma²) | Three-fold generate | | (311) (131) (113) | (11h²/8ma²) | Three-fold generate etc. | ## Operators: - An operator is a short hand notation for a set of well-defined mathematical operations to be carried out on a function. - The function on which the operator operates is called an operand. - Examples of operators are: *d/dx*, *d/dy*, *d/dz*. ## Rules of operators: 1. **Addition and subtraction of operators:** If *Â* and *B^ * are two operators, the addition or subtraction of these operators on a function *f(x)* is defined as: *(A+B) f(x) = A f(x) + B f(x)* and *(A - B) f(x) = A f(x) - B f(x)*. 2. **Product of operators:** The product of two operators *A* and *B^ * on a function *f(x)* is defined as: *Â B f(x) = Â [B f (x)]*. In this case, *B* is first operated on the function *f(x)*, then *A* is operated on the resulting function. 3. **Commentator of operators:** If *A* and *B* are two operators, the product *A B* may be the same as *B A*. If they are not the same, *AB = BÂ; the commutator of the two operators is defined as: *[A, B] = A B - B A ≠ 0*. 4. **Linear operators:** Linear operators are the operators for which the following conditions hold true: *Â [f(x) + g(x)] = A f(x) + g(x)* and *A [c.f(x)] = c. A f(x)*, where *f(x)* and *g(x)* are two functions, A_ is an operator and *c* is a constant. - All quantum mechanical operators are linear. ## Quantum mechanical operator: - A quantum mechanical operator is a short hand notation for well-defined mathematical operations to be carried out on a function. ## Hermitian operator: - An operator *Â* is said to be Hermitian if: *∫ψ₁*^(Âψ₂)*dτ = ∫(Âψ₁)*ψ₂*dτ* - *ψ₁* and *ψ₂* are the Eigen functions of the operator *A* and *ψ₁*** is the complex conjugate of *ψ₁*. - All quantum operators are Hermitian. ## Ladder operators: - Ladder operators are also known as shift operators, step-up and step-down operators, or raising and lowering operators. - They are defined as: *J_+ = J_x + iJ_y* and *J_- =J_x + iJ_y*. ## Complex conjugate: - If *A* and *B* are two real functions and the wave function *ψ* is defined as *ψ = A + iB*, where *i* is an imaginary number (*i² = -1*), then the complex conjugate of *ψ* is: *ψ** = A - iB*. - The absolute value of *ψ* is given by: *|ψ|² = ψ*ψ = A²- i²B² = A² + B²*. - The square of the absolute magnitude of the wave function ψ, *|ψ|²*, is always a positive quantity. ## Numerical problem: 1. **Calculate the energies of an electron confined to move in a one-dimensional box of width 1A°.** - Using the equation for energy in a 1D box (E = n²h²/8ma²). - If *a = 1A° = 10⁻¹⁰ m*, *h = 6.626 × 10⁻³⁴ Js*, and *m = 9.1 × 10⁻³¹ kg*, then *E = (n²(6.626 ×10⁻³⁴ J.s)²)/8×9.1×10⁻³¹ ×(10⁻¹⁰)2 x (1.602×10⁻¹⁹) = 37.6 n² eV ≈ 38 n² eV*. - Therefore, the energies of the electron corresponding to different values of *n* are: - For *n = 1: E₁ = 30 eV*. - For *n = 2: E₂ = 38 ×(2)²= 152 eV*. - For *n = 3: E₃ = 38 ×(3)² = 342 eV*. - For *n = 4: E₄ = 38 ×(4)² = 608 eV etc*. 2. **Determine degeneracy of the energy level (14h²/8ma²) of a particle in the cubical box.** - Using the equation for the energy level of the particle in a cubical box: *E = (h²/8ma²)(n_x² + n_y² + n_z²)* - Setting *E = (14h²/8ma²)* we get (n_x² + n_y² + n_z²) = 14. - The ways in which the sum of these squares will be 14 are: - n_x = 1, n_y = 2, n_z = 3 - n_x = 1, n_y = 3, n_z = 2 - n_x = 2, n_y = 1, n_z = 3 - n_x = 2, n_y = 3, n_z = 1 - n_x = 3, n_y = 1, n_z = 2 - n_x = 3, n_y = 2, n_z = 1 - Therefore, the energy level is sixfold degenerate. ## Multiple Choice Questions: 1. Quantum mechanics is based on a) de Broglie hypothesis c) Planck’s quantum theory **d) All of these** 2. The correct form of Schrodinger wave equation for a particle in one-dimensional box is: **a) (d²/dx²) + (8π²m/h²)(E - P.E)ψ = 0** b) (d²/dx²) + (8π²m/h²)(E + P.E)ψ = 0 c) (d²/dx²) - (8π²m/h²)(E- P.E)ψ = 0 d) (d²/dx²) - (8π²m/h²)(E + P.E)ψ = 0 3. The correct expression for energy of an electron in one dimensional box is: a) E_n = (n²h²/4ma²) b) E_n = (n²h²/2ma²) **c) E_n = (n²h²/8ma²)** d) E_n = (n²h²/mla²) 4. de Broglie's hypothesis hold good for: **a) only microscopic particles** b) only macroscopic particles c) Both (a) and (b) d) Cannot be predicted. 5. Which of the following represents the correct form of Hamiltonian operator? a) (h²/2m)∇² - V b) (h²/2m)∇² - V **c) (h²/2m)∇² + V** d) None of these 6. “Momentum of a particle in motion is inversely proportional to wavelength.” This is the statement of: a) Heisenberg’s uncertainty principle **b) de Broglie hypothesis** c) Aufbau principle d) Planck radiation law 7. “It is impossible to determine accurately or precisely both the position and momentum of a moving microscopic particle simultaneously” is the statement of: **a) Heisenberg's uncertainty principle** b) de Broglie hypothesis c) Aufbau principle d) Planck radiation law 8. According to the de Broglie’s hypothesis, the momentum of a particle is _________________________ proportional to wavelength. a) Directly **b) Inversely** c) Remains the same d) Cannot be predicted. 9. The de Broglie's equation can be expressed as: **a) λ = h / mv** b) λ = mv / h c) λ = h² / mv d) λ = h / m²v² 10. The equation *Δp × Δx ≥ h/4* represents: a) De Broglie hypothesis **b) Heisenberg’s uncertainty principle** c) Wave function d) Planck’s radiation law 11. The symbol (ψ) in the Schrodinger wave equation represents: a) Wavelength of spherical wave **b) Amplitude of spherical wave** c) Frequency of spherical eave d) None of these