6 ATOMIC ORBITAL THEORY 1.10 FUNDAMENTAL WAVE (OR FIRST HARMONIC) The stationary wave with no nodal point other than the fixed ends.It is the wave from which the frequency 'of all other waves in a set is generated by multiplying the fundamental frequency vby an integer n: v'=nv (1.10) Example.In the fundamental wave,in Fig.1.10,the amplitude may be consid ered to be oriented upward and to continuously increase from either fixed end,reach- ing a maximum at the midpoint.In this"well-behaved"wave,the amplitude is zero at each end and a maximum at the center. 12 Figure 1.10.The fundamental wave 1.11 FIRST OVERTONE (OR SECOND HARMONIC) The stationary wave w vith one nodal point located at the midpoint(n=2 in the equa tion given in Sect.1.10).It has half the wavelength and twice the frequency of the first harmonic. Example.In the first overtone (Fig.1.11),the nodes are located at the ends and a the point half-way between the ends,at which point the amplitude changes direction The two equal segments of the wave are portions of a single wave;they are not inde pendent.The two maximum amplitudes come at exactly equal distances from the ends but are of opposite signs. 1.12 MOMENTUM (P) This is the vectorial property (i.e.,having both magnitude and direction)of a mov- ing particle:it is equal to the mass m of the particle times its velocity v: p=mv (1.12)
1.10 FUNDAMENTAL WAVE (OR FIRST HARMONIC) The stationary wave with no nodal point other than the fixed ends. It is the wave from which the frequency ν of all other waves in a set is generated by multiplying the fundamental frequency ν by an integer n: ν nν (1.10) Example. In the fundamental wave, λ/2 in Fig. 1.10, the amplitude may be considered to be oriented upward and to continuously increase from either fixed end, reaching a maximum at the midpoint. In this “well-behaved” wave, the amplitude is zero at each end and a maximum at the center. 1.11 FIRST OVERTONE (OR SECOND HARMONIC) The stationary wave with one nodal point located at the midpoint (n 2 in the equation given in Sect. 1.10). It has half the wavelength and twice the frequency of the first harmonic. Example. In the first overtone (Fig. 1.11), the nodes are located at the ends and at the point half-way between the ends, at which point the amplitude changes direction. The two equal segments of the wave are portions of a single wave; they are not independent. The two maximum amplitudes come at exactly equal distances from the ends but are of opposite signs. 1.12 MOMENTUM (P) This is the vectorial property (i.e., having both magnitude and direction) of a moving particle; it is equal to the mass m of the particle times its velocity v: p mv (1.12) 6 ATOMIC ORBITAL THEORY 1/2 λ Figure 1.10. The fundamental wave. c01.qxd 5/17/2005 5:12 PM Page 6
ORBITAL (ATOMIC ORBITAL)7 nodal point Figure 1.11.The first overtone (or second harmonic)of the fundamental wave. 1.13 DUALITY OF ELECTRONIC BEHAVIOR Particles of small mass such as may exhibit prop erties of either particles (they have momentum)or waves(they can be defracted like light waves).A single experiment may demonstrate either particle properties or wave properties of elec- trons,but not both simultaneously. 1.14 DE BROGLIE RELATIONSHIP The wavelengh()by by Louis de Broglie(1892-1960): 入=hlp=hlmw (1.14) where h is Planck's constant,m the mass of the particle,and v its velocity.This rela- tionship makes it possible to relate the momentump of the electron,a particle prop- erty,.with its wavelength，a wave property.. 1.15 ORBITAL (ATOMIC ORBITAL) A wave description of the size,shape,and orientation of the region in space avail- able to an electron;each orbital has a specific energy.The position (actually the probability amplitude)of the electron is defined by its coordinates in space,which in Cartesian coordinates is indicated by w(x,y,).W cannot be measured directly:it is a mathematical tool.In terms of spherical coordinates,frequently u sed in calcula tions,the wave function is indi ated h )where(Fig.1.5)is the radial distance ofa point from theangle between the radial lin and the
1.13 DUALITY OF ELECTRONIC BEHAVIOR Particles of small mass such as electrons may exhibit properties of either particles (they have momentum) or waves (they can be defracted like light waves). A single experiment may demonstrate either particle properties or wave properties of electrons, but not both simultaneously. 1.14 DE BROGLIE RELATIONSHIP The wavelength of a particle (an electron) is determined by the equation formulated by Louis de Broglie (1892–1960): λ h/p h/mv (1.14) where h is Planck’s constant, m the mass of the particle, and v its velocity. This relationship makes it possible to relate the momentum p of the electron, a particle property, with its wavelength λ, a wave property. 1.15 ORBITAL (ATOMIC ORBITAL) A wave description of the size, shape, and orientation of the region in space available to an electron; each orbital has a specific energy. The position (actually the probability amplitude) of the electron is defined by its coordinates in space, which in Cartesian coordinates is indicated by ψ(x, y, z). ψ cannot be measured directly; it is a mathematical tool. In terms of spherical coordinates, frequently used in calculations, the wave function is indicated by ψ(r, θ, ϕ), where r (Fig. 1.15) is the radial distance of a point from the origin, θ is the angle between the radial line and the ORBITAL (ATOMIC ORBITAL) 7 nodal point λ Figure 1.11. The first overtone (or second harmonic) of the fundamental wave. c01.qxd 5/17/2005 5:12 PM Page 7
8 ATOMIC ORBITAL THEORY z-axis,and is the angle between the x-axis and the projection of the radial line on the xy-plane.The relationship between the two coordinate systems is shown in Fig.1.15.An orbital centered on a single atom (an atomic orbital)is frequently denoted as(phi)rather than w(psi)to distinguish it from an orbital centered on more than one atom (a molecular orbital)that is almost always designated w. The projection ofon the =OB.and OBA is a righ tangle.Hence cos =z/r,and th oC,but OC= r sin 0.Hence. rsin coso.Similarly,sin=ylAB:therefore,y=AB sin sin0sin Accordingly,a point(x.y.z)in Cartesian coordinates is transformed to the spherical coordinate system by the following relationships: 7=re0e日 y=r sine sin x=r sin 0 cos &9gpaeelaoent Origin (0) Figure 1.15.The relationship between Cartesian and polar coordinate systems. 1.16 WAVE FUNCTION In quantum mechanics,the wave function is synonymous with an orbital
z-axis, and ϕ is the angle between the x-axis and the projection of the radial line on the xy-plane. The relationship between the two coordinate systems is shown in Fig. 1.15. An orbital centered on a single atom (an atomic orbital) is frequently denoted as φ (phi) rather than ψ (psi) to distinguish it from an orbital centered on more than one atom (a molecular orbital) that is almost always designated ψ. The projection of r on the z-axis is z OB, and OBA is a right angle. Hence, cos θ z/r, and thus, z r cos θ. Cosϕ x/OC, but OC AB r sin θ. Hence, x r sin θ cos ϕ. Similarly, sin ϕ y/AB; therefore, y AB sin ϕ r sin θ sin ϕ. Accordingly, a point (x, y, z) in Cartesian coordinates is transformed to the spherical coordinate system by the following relationships: z r cos θ y r sin θ sinϕ x r sin θ cosϕ 1.16 WAVE FUNCTION In quantum mechanics, the wave function is synonymous with an orbital. 8 ATOMIC ORBITAL THEORY Z x y z r θ φ φ θ Origin (0) volume element of space (dτ) B A Y X C Figure 1.15. The relationship between Cartesian and polar coordinate systems. c01.qxd 5/17/2005 5:12 PM Page 8
PROBABILITY INTERPRETATION OF THE WAVE FUNCTION 9 1.17 WAVE EQUATION IN ONE DIMENSION The mathematical description of an orbital involving the amplitude behavior of a wave.In the case of a one-dimensional standing wave,this is a second-order differ- ential equation with respect to the amplitude: dfx)/dx2+(4π23fx)=0 (1.17) whereis the wavelength and the amplitude function isf(). 1.18 WAVE EQUATION IN THREE DIMENSIONS The function f(x,y.z)for the wave equation in three dimensions,analogous to f(x). which describes the amplitude behavior of the one-dimensional wave.Thus,f(x,y.z) satisfies the equation af(x)/ax2+af(y)/ay2+af(z)/az2+()f(x.y.z)=0 (1.18) In the portion aaxis an operator that says"partially dif- ferentiate twice with respect to x that which follows. 1.19 LAPLACIAN OPERATOR ect to the three Cartesian es in Eq. 1.18is called the operator (after Pierre S.Laplace 2(del squared) V2 =a21ax2+a2/ay2+a21a2 (1.19a) which then simplifies Eq.1.18 to xy,)+(4π2n3fxy,)=0 (1.19b) 1.20 PROBABILITY INTERPRETATION OF THE WAVE FUNCTION The wave function (or orbital)(r).because it is related to the amplitude of a wave that determines the location of the electron,can have either negative or positive val- ues.However,a probability,by definition,must always be positive,and in the pres ent case this can be achieved by qangeaplmdcAceoiney,thePoab西 electro in a speci ele ent of sp ce dr nce r from nucleus is r)dt.Although the orbital,has mathematical significance (in
1.17 WAVE EQUATION IN ONE DIMENSION The mathematical description of an orbital involving the amplitude behavior of a wave. In the case of a one-dimensional standing wave, this is a second-order differential equation with respect to the amplitude: d 2 f(x)/dx2 (4π 2 /λ2 ) f(x) 0 (1.17) where λ is the wavelength and the amplitude function is f(x). 1.18 WAVE EQUATION IN THREE DIMENSIONS The function f(x, y, z) for the wave equation in three dimensions, analogous to f(x), which describes the amplitude behavior of the one-dimensional wave. Thus, f(x, y, z) satisfies the equation 2 f(x)/x2 2 f(y)/y2 2 f(z)/z2 (4π2 /λ2 ) f(x, y, z) 0 (1.18) In the expression 2 f(x)/x 2 , the portion 2 /x 2 is an operator that says “partially differentiate twice with respect to x that which follows.” 1.19 LAPLACIAN OPERATOR The sum of the second-order differential operators with respect to the three Cartesian coordinates in Eq. 1.18 is called the Laplacian operator (after Pierre S. Laplace, 1749–1827), and it is denoted as ∇2 (del squared): ∇2 2 /x2 2 /y2 2 /z2 (1.19a) which then simplifies Eq. 1.18 to ∇2 f(x, y, z) (4π2 /λ2 ) f(x, y, z) 0 (1.19b) 1.20 PROBABILITY INTERPRETATION OF THE WAVE FUNCTION The wave function (or orbital) ψ(r), because it is related to the amplitude of a wave that determines the location of the electron, can have either negative or positive values. However, a probability, by definition, must always be positive, and in the present case this can be achieved by squaring the amplitude. Accordingly, the probability of finding an electron in a specific volume element of space dτ at a distance r from the nucleus is ψ(r)2 dτ. Although ψ, the orbital, has mathematical significance (in PROBABILITY INTERPRETATION OF THE WAVE FUNCTION 9 c01.qxd 5/17/2005 5:12 PM Page 9
10 ATOMIC ORBITAL THEORY that it can have negative and positive values).2has physical significance and is always positive. L21 SCHRODINGER EOUATION This is a differential equation,formulated by Erwin Schrodinger (1887-1961) whose solution is the w for the m under co tion takest nsideration.Thisequ orm an equation fo andin Itis fron this for the term wave mechan of the io toa wave cquation (Sect.1.18)isdem stituting the de Broglie equation(1.14)into Eq.1.19b and replacing fby 720+(4π2m21v2h2b=0 (1.21a) To incorporate the total energy Eof a electron into this equation,use is made of the fact that the total energy is the sum of the potential energy V,plus the kinetic energy, 1/2m2,or 2=2(E-V/m (1.21b) Substituting Eq.1.21b into Eq.1.21a gives Eq.1.21c: 720+(8π2mlh2(E-V)0=0 (1.21c) which is the Schrodinger equation. 1.22 EIGENFUNCTION This is a hybrid German-English word that in English might be translated as"char acte on”;itis n,which can solutions of the wave equation,Eq.1.17 [e.g.,fx)must be zero at each end,as in the case of the vibrating string fixed at both ends;this is the so-called boundary condi- tion].In general,whenever some mathematical operation is done on a function and the same function is regenerated multiplied by a constant,the function is an eigen- function,and the constant is an eigenvalue.Thus,wave Eq.1.17 may be written as df)/d2=-(4π2n3)fx (1.22 This equation is an eigenvalue equation of the form: (Operator)(eigenfunction)=(eigenvalue)(eigenfunction)
that it can have negative and positive values), ψ2 has physical significance and is always positive. 1.21 SCHRÖDINGER EQUATION This is a differential equation, formulated by Erwin Schrödinger (1887–1961), whose solution is the wave function for the system under consideration. This equation takes the same form as an equation for a standing wave. It is from this form of the equation that the term wave mechanics is derived. The similarity of the Schrödinger equation to a wave equation (Sect. 1.18) is demonstrated by first substituting the de Broglie equation (1.14) into Eq. 1.19b and replacing f by φ: ∇2 φ (4π2 m2 v2 /h2 )φ 0 (1.21a) To incorporate the total energy E of an electron into this equation, use is made of the fact that the total energy is the sum of the potential energy V, plus the kinetic energy, 1/2 mv2 , or v2 2(E V)/m (1.21b) Substituting Eq. 1.21b into Eq. 1.21a gives Eq. 1.21c: ∇2 φ (8π2 m/h2 )(E V)φ 0 (1.21c) which is the Schrödinger equation. 1.22 EIGENFUNCTION This is a hybrid German-English word that in English might be translated as “characteristic function”; it is an acceptable solution of the wave equation, which can be an orbital. There are certain conditions that must be fulfilled to obtain “acceptable” solutions of the wave equation, Eq. 1.17 [e.g., f(x) must be zero at each end, as in the case of the vibrating string fixed at both ends; this is the so-called boundary condition]. In general, whenever some mathematical operation is done on a function and the same function is regenerated multiplied by a constant, the function is an eigenfunction, and the constant is an eigenvalue. Thus, wave Eq. 1.17 may be written as d2 f(x)/dx2 (4π2 /λ2 ) f(x) (1.22) This equation is an eigenvalue equation of the form: (Operator) (eigenfunction) (eigenvalue) (eigenfunction) 10 ATOMIC ORBITAL THEORY c01.qxd 5/17/2005 5:12 PM Page 10