1
Second-Quantization and Diagrammatic Many-Body Perturbation Theory 2nd Year Masters, Theoretical Chemistry Mark E. Casida Date de publication: January 27, 2010
Richard Feynman bought a van in 1975. This van came to be covered with his now famous diagrams. Naturally, few people would have recognized these strange pictures at the time. However it seems that Feynman was once asked why he had Feynman diagrams on his van. His answer was simple: “Because I am Richard Feynman.” The van’s licence plate was “QANTUM” (California license plates were limited to 6 letters.)
2
Contents I
Perturbation Theory
7
1 Time-Independent Theory 11 1.1 Rayleigh-Schr¨odinger Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Time-Dependent Theory 2.1 Adiabatic Perturbation Theory 2.2 Pictures . . . . . . . . . . . . . 2.3 Evolution Operator . . . . . . . 2.4 Linked Cluster Theorem . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
3 Appendices 3.1 Units . . . . . . . . . . . . . . . . . . . . 3.2 Proof of the 2n + 1 Rule . . . . . . . . . 3.3 Second-Order RSPT Wave Function . . . 3.4 Fourier Transforms . . . . . . . . . . . . 3.5 The Dirac δ Function . . . . . . . . . . . 3.6 First-Order RSPT Wave Function . . . . 3.7 Third-Order RSPT Wave Function . . . 3.8 Rules for One-Particle Energy Diagrams
II
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . .
19 19 21 23 25
. . . . . . . .
29 29 31 33 34 36 37 39 40
Many-Body Perturbation Theory
4 Time-Independent Theory 4.1 Creation and Annihilation Operators 4.2 Second-Quantized Operators . . . . . 4.3 Density Matrices . . . . . . . . . . . 4.4 Particle-Hole Representation . . . . . 4.5 Wick’s Theorem (simplified version) . 4.6 Hartree-Fock Energy Expression . . . 4.7 Brillouin’s Theorem . . . . . . . . . . 4.8 CIS . . . . . . . . . . . . . . . . . . . 4.9 MP2 . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
43 . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
47 47 50 51 52 54 55 57 57 60
5 Time-Dependent Theory 63 5.1 Wick’s Theorem (full version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3
4
CONTENTS 5.3
Summary of Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Appendices 6.1 Evaluation of hΦ|ˆ v|Φai i Using Wick’s Theorem v |Φbj i Using Wick’s Theorem 6.2 Evaluation of hΦai |ˆ 6.3 Evaluation of hΦ|ˆ v|Φab ij i Using Wick’s Theorem 6.4 Rules for Goldstone Energy Diagrams . . . . . 6.5 Rules for Hugenholtz Energy Diagrams . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
67 69 69 71 76 78 78
Preface This is my 3rd year of teaching courses for the French Label National de Chimie Th´eorique. This teaching is done with a willing heart, in the belief that France needs well-trained students in theoretical chemistry. During my first 2 years of teaching for the Label I focused on teaching my specialty, namely timedependent density-functional theory (TDDFT). However the Label prizes diversity, so I have decided to return to another area in which I have enjoyed dabbling from time to time [1, 2, 3, 4, 5, 6, 7, 8] and share my expertise with many-body perturbation theory (MBPT) — an area not completely unrelated to TDDFT [8]. The course focuses on Goldstone’s approach [9] which I think is most easily accessible to students new to MBPT. The course divides naturally into two parts. The first is ordinary perturbation theory. This serves as a review of Rayleigh-Schr¨odinger perturbation theory, of time-dependent perturbation theory, and as a first introduction to diagrams. The second part of the course focuses on MBPT. The goal is not to be exhaustive, but to introduce second quantization, Wick’s theorem in all its glory, and then finally to justify the diagrams introduced in the first part of the course. Most sections begin with random phrases in italics. These are just short thoughts that entered my mind when writing the section. Perhaps they will succeed in communicating concisely my inner feelings about each chapter. Please take them in the spirit they are offered—with a touch of whimsy. I expect students to follow the course and to participate by deriving some of the expressions as we go. Suitable exercises have been included in these course notes. Answers appear in appendices. What I hope students will learn from the course is how to write appropriate diagrams for the perturbation expressions they develop, rather than to become overdependent on diagrams. In this context the diagrams serve as mneumonics and as checks on algebra. This is the way I work and I find it to be very useful. In past years I have tried to provide bilingual notes in English and French. For now, the notes for this course are just in English, making them useful to the increasingly large number of international students coming to France. As to French students, they are increasingly proficient in this lingua franca of science and often eager to become still more proficient. Pourtant la langue d’instruction du cours sera le fran¸cais sauf raison de faire autrement. Mark E. Casida 4 January 2010 Grenoble, France
5
6
CONTENTS
Part I Perturbation Theory
7
9 In the beginning there was one. This course focuses on solving the Schr¨odinger equation for N electrons. There are two ways to look at such a system. We may either consider the N electrons as a single (complicated) body, in which case we are doing ordinary perturbation theory. Or we may chose to take the many-body aspects of the N electron system explicitly into account, in which case we are doing many-body perturbation theory (MBPT). This introductory chapter focuses on ordinary perturbation theory because this should already be at least somewhat familiar to the student who will then hopefully find it all the easier to assimilate new unfamiliar concepts, such as diagrams, as they are gradually introduced. The most basic Schr¨odinger equation is the time-dependent form, d ˆ HΨ(t) = i Ψ(t) , dt
(1)
written here in atomic units (e = me = h ¯ = 1, Appendix 3.1). Notice that we are following the common convention that upper-case quantities (operators, wave functions, etc.) refer to N-electron quantities, while lower-case quantities are typically reserved for one-electron quantities (e.g., spinorbitals and their corresponding energies). The time-dependent Schr¨odinger equation may be more familiar to physicists (who often want to describe scattering experiments) than to chemists. Chemists usually work with the time-independent Schr¨odinger equation which is obtained by making the well-known stationary state ansatz that the wave function, Ψ(t), can be written as a product of a spatial part and of a time-dependent part. It is then shown in standard elementary textbooks that, ˆ I = EI ΨI , HΨ
(2)
Ψ(t) = ΨI e−iEI t .
(3)
E0 ≤ E1 ≤ E2 ≤ E3 ≤ · · · ,
(4)
and The indices are such that, but we will usually assume strict inequalities. That is, we will usually restrict ourselves to nondegenerate perturbation theory. We will begin by recalling Rayleigh-Schr¨odinger perturbation theory (RSPT) for the time-independent case. This will also be an excuse to introduce the idea of diagrams. However we will not “derive” the diagrams until Part II as their origin is heavily rooted in second-quantization and Wick’s theorem. Time-dependent perturbation theory will also be reviewed in Part I.
10
Chapter 1 Time-Independent Theory 1.1
Rayleigh-Schr¨ odinger Perturbation Theory It was in the order of things...
It is difficult to imagine reaching the 2nd year of Masters’ study in Physical Chemistry/Chemical Physics without having seen at least a very basic presentation of Rayleigh-Schr¨odinger perturbration theory (RSPT). We treat it again so that we can focus on something new, namely diagrams, in a context which should already be a bit familiar. Our perturbed problem, ˆ I = EI ΨI , HΨ (1.1) resembles an unperturbed problem,
We can write,
(0) (0) ˆ (0) Ψ(0) H I = EI ΨI .
(1.2)
ˆ =H ˆ (0) + λVˆ , H
(1.3)
ˆ −H ˆ (0) . Vˆ = H
(1.4)
where,
The parameter λ = 1 is formal and is used to define the notion of the order of the perturbation. As a general rule perturbations divide into two types, namely physical pertubations and formal perturbations. What has just been described is a formal perturbation because we have no way to control the size of the perturbation and force it to be small—we can only try and see how it works. In contrast, physical perturbations arise for physical reasons such as applied electromagnetic fields or distortions of molecular geometries. Their strength (λ) can be controlled and perturbation theory will always work when the perturbation is small enough. Among other things perturbation theory in this context provides a standard way to evaluate analytic derivatives, such as forces on nuclei for automatic geometry optimizations. It is also used in nonlinear optics to evalute polarizabilities and in nuclear magnetic resonance (NMR) to evaluate chemical shifts. For now, we will not worry about the exact nature of the perturbation. We will just develop the theory and see if it is useful later. We want to solve the equation,
ˆ (0) + λVˆ H
=
(0)
(1)
(0)
(1)
(2)
ΨI + λΨI λ2 ΨI + · · ·
(2)
EI + λEI + λ2 EI + · · · 11
(0)
(1)
(2)
ΨI + λΨI λ2 ΨI + · · · .
(1.5)
12
CHAPTER 1. TIME-INDEPENDENT THEORY
Do the multiplations and use the linear independence of the λn monomials to determine the orderby-order perturbation equations. λ0 Coefficient: ˆ (0) Ψ(0) = E (0) Ψ(0) H (1.6) I I I This is just the unperturbed equation! λ1 Coefficient: ˆ (0) Ψ(1) + Vˆ Ψ(0) = E (0) Ψ(1) + E (1) Ψ(0) H I I I I I I λ2 Coefficient:
(0) (2) (1) (1) (2) (0) ˆ (0) Ψ(2) ˆ (1) H I + V ΨI = EI ΨI + EI ΨI + EI ΨI
··· λn Coefficient: ˆ (0) Ψ(n) ˆ (n−1) = H I + V ΨI
n X
(k)
(n−k)
EI ΨI
(1.7)
(1.8)
(1.9)
k=0
We will solve these equations subject to two conditions: 1. Orthonormality
(0)
(0)
hI (0) |J (0) i = hΨI |ΨJ i = δI,J 2. Intermediate normalization
(0)
hI (0) |Ii = hΨI |ΨI i = 1
(1.10) (1.11)
A consequence of these conditions is that the wave function is no longer normalized, hI|Ii = 6 1
(1.12)
(although sometimes hI|Ii = 1 is imposed instead of the intermediate normalization [10].) We have, in place of the normal normalization, that, 1 = hI (0) |Ii = hI (0) |I (0) i + λhI (0) |I (1) i + λ2 hI (0) |I (2) i + · · · ,
(1.13)
hI (0) |I (n) i = δ0,n .
(1.14)
so that, An immediate consequence follows from applying Eq. (1.14) to Eq. (1.9), ˆ (0) |I (n) i + hI (0) |Vˆ |I (n−1) i = hI (0) |H Thus,
and
n X
k=0
(k)
EI hI (0) |I (n−k) i
(n) (0) EI δn,0 +hI (0) |Vˆ |I (n−1) i = EI , {z } zero
|
(n)
EI
(1)
EI
= hI (0) |Vˆ |I (n−1) i ⇒ = hI (0) |Vˆ |I (0) i .
(1.15)
(1.16)
(1.17)
¨ 1.1. RAYLEIGH-SCHRODINGER PERTURBATION THEORY
13
So the (n − 1)st correction to the wave function suffices to determine the nth order correction to the energy. Most textbooks stop with the first-order correction for Ψ: Theorem. (K6=I) (0) ˆ (0) X |V |I i (1) (0) hK (1.18) ΨI = ΨK (0) (0) EI − EK K=1,∞ so (K6=I) ˆ |I (0) i|2 X |hK (0) |V (2) EI = (1.19) (0) (0) EI − EK K=1,∞ Proof. The intermediate normalization condition, hI (0) |I (1) i = 0 ⇒
(K6=I)
(1)
ΨI
=
X
(0)
(1)
ΨK CK,I .
(1.20)
K=1,∞
Let us try to find the coefficients,
(1)
CK,I = hK (0) |I (1) i .
(1.21)
Multiplication followed by integration (“multigration”) of the first-order perturbation equation [Eq. (1.7)]
(0)∗
by ΨK
(0) (1) (1) (0) ˆ (0) Ψ(1) ˆ (1) H I + V ΨI = EI ΨI + EI ΨI
(1.22)
ˆ (0) |I (1) i +hK (0) |Vˆ |I (0) i = E (0) hK (0) |I (1) i +E (1) hK (0) |I (0) i . hK (0) |H I I | | {z } {z } | {z } (0) (1) zero E hK (0) ||I (1) i C
(1.23)
give us for K 6= I,
|
K
{z
K,I
}
(0) (1) EK CK,I
So
(0)
(0)
EI − EK
(1)
CK,I (1)
CK,I (1)
ΨI ♥
= hK (0) |Vˆ |I (0) i ⇒ hK (0) |Vˆ |I (0) i = (0) (0) EI − EK ⇒ ∞ (0) ˆ (0) X |V |I i (0) hK . = ΨK (0) (0) EI − EK K=1,∞
(1.24)
We have already seen the very useful relation that, (n)
EI
= hI (0) |Vˆ |I (n−1) i ,
However practical calculations can often profit from the Theorem (2n + 1 rule)
(1.25)
14
CHAPTER 1. TIME-INDEPENDENT THEORY
The wave functions through order n determine the energy through order 2n + 1. Precise formula using the intermediate normalization are [10], (2n)
EI
(2n+1)
EI
= hI (n−1) |Vˆ |I (n) i − = hI (n) |Vˆ |I (n) i −
n−1 X 2n−k−1 X k=1 j=n−k
n X
2n−k X
(j)
EI hI (k) |I (2n−k−j)i (j)
k=1 j=n−k+1
EI hI (k) |I (2n−k−j+1)i .
(1.26)
Exercise: The 2n + 1 rule is a special case of the more general rule, (n)
EI
= hI (m) |Vˆ |I (n−m−1) i −
m X
n−k−1 X
(j)
hI (k) |I (n−k−j)iEI +
k=1 j=m−k+1
m X
(n)
δn,k EI .
(1.27)
k=1
This rule is trivial for m = 0. The general tricks needed to derive Eq. (1.27) may be found in Ref. [11] pp. 252-253. Use the two relations, (n−1) Vˆ ΨI =
(0) ˆ (0) Ψ(n) + EI − H I
(n−1) (0) ˆ (0) Ψ(n) = Vˆ ΨI − EI − H I
n X
(n−j)
ΨI
n X
(n−j)
ΨI
(j)
EI
j=1 (j)
EI ,
(1.28)
j=1
which follow from Eq. (1.9) to prove Eq. (1.27) for the cases m = 1, 2, 3 and then deduce the general formula. The solution is given in Appendix 3.2. Note that the case with m = 1 is sufficiently important that it is worth deriving it oneself just to understand its origin.
1.2
Diagrams A picture is worth a thousand words.
Even RSPT rapidly becomes complicated as we go to higher and higher order. What is needed is some type of bookkeeping device that will help us to keep track of terms and to avoid making mistakes. These are the famous “Feynman diagrams.” (We are not yet in a position to distinguish between Feynman diagrams and other closely-related diagrams. Once we are in this position, we will see that the diagrams in this section are not exactly what we will be calling Feynman diagrams later.) Before describing the diagrams, it is worth making the distinction between “translating” and “enumerating.” • Translating is the easier of the two operations: It consists of writing the diagram corresponding to a given perturbation theory formula—that is, translating the formula into a diagram. This is useful because the ability to draw the diagram provides both a check on the validity of the formula but also on the completeness of the ensemble of diagrams. • Enumerating is the more difficult of the two operations: It consists of writing down each possible diagram exactly once and then translating them into correct formulae. Not only does one have to know the rules for backtranslating diagrams into formula quite well, but it is necessary to be able to recognize when two or more diagrams are equivalent (and hence redundant), when diagrams are impossible (e.g., unlinked diagrams are impossible), and when possible diagrams are missing.
1.2. DIAGRAMS
15
Beginners often try to begin with enumeration and translation into formulae, rather than deriving formulae and then translating them into diagrams. The result is at best an overreliance on diagrams and at worst an overabundance of na¨ıve errors rendering calculations useless. This is why I emphasize translating more than enumerating. Behind the diagrams are a series of concepts (some of which will be at least a little mysterious until we do full time-dependent MBPT.) The first is the particle-hole picture which might be illustrated as,
In order to define the particle-hole concept, we need to first define a reference state, known as the (0) physical vaccuum. For concreteness, we will take this as the ground state, Ψ0 . Exciting to the (0) (0) (0) Ith state, ΨI , is described as making a “particle” in ΨI and a “hole” in Ψ0 . These concepts are awkward to apply when thinking about the ensemble of N electrons as a single (complicated) body. They make more sense when thinking about single electron systems. For such a system, we can remove an electron to make a “hole” in the ground state and add an electron to make a “particle” above the ground state or induce “particle-hole” states which are simply excitations. Closely associated with this is the representation of states as arrows moving forward or backward in time,
Holes move backward in time, while particles move forward in time. Since we have not yet talked about time-dependent theory you may well ask yourself how time enters into this theory. Save that question. We will get to its answer–just not right away. The final element that we will need to draw diagrams is the matrix element,
16
CHAPTER 1. TIME-INDEPENDENT THEORY
The important thing here is that indices for incoming lines go on the right of the matrix element and indices for outgoing lines go on the left. We can now write the diagram corresponding to the formula, (I6=0)
(1)
Ψ0 =
X
I=1,∞
(0) hI
ΨI
(0)
|Vˆ |0(0) i
(0)
(0)
E0 − EI
.
(1.29)
namely
The usual way to write the diagram is as in part (a). Note that the diagram could equivalently have been drawn with the incoming arrow on the left and the outgoing arrow on the right. Only the diagram topology is important. In part (b), the particle and hole lines have been labeled, so that the matrix element hI (0) |Vˆ |0(0) i is clear. A horizontal “time cut” has been indicated by a solid line. This (0) (0) represents the denominator, E0 − EI . In a general diagram, there is one time cut between each P P V -circle. Each time cut represents a denominator of the form hole energies − particle energies. A dotted “capping line” has also been added to diagram (b). This is needed to determine the sign in front of the sum, sign = (−1)h+l . (1.30) Here h is the number of hole lines (h = 1) and l is the number of loops (l = 1). Overall the sign of this diagram is +1. In the case of wave function diagrams, there is an implicit sum over the (0) excited-state functions (ΨI in this case.) It is now easy to use the formula, (n+1)
E0
= h0(0) |Vˆ |0(n) i ,
to obtain the second-order energy diagrams from those for the first-order wave function,
(1.31)
17
1.2. DIAGRAMS The corresponding formula is, (2)
E0 =
X
I=1,∞
h0(0) |Vˆ |I (0) ihI (0) |Vˆ |0(0) i (0)
(0)
E0 − EI
.
(1.32)
Notice that the diagram implies a sum over the index I. The first-order energy diagram is,
There is a single hole loop but no time cuts are possible. The corresponding formula is, (1)
E0 = h0(0) |Vˆ |0(0) i .
(1.33)
Exercise: Derive expressions for the second-order part of the RSPT wave function and show that they can be represented by the diagrams,
Notice that these are just all possible time orderings of a single diagram. Give the corresponding third-order energy expressions and diagrams. The solution is given in Appendix 3.3. It may be interesting to think about the diagrammatic interpretation of the 2n + 1 rule.
18
CHAPTER 1. TIME-INDEPENDENT THEORY
Chapter 2 Time-Dependent Theory The more things change, the more they stay the same. We will now leave the old familiar Rayleigh-Schr¨odinger perturbation theory and stray into the time domain. There are several reasons for doing so. One is the fascinating area of nonlinear optics (NLO) which requires being able to handle electric fields that oscillate in time — that is, to be able to treat the classical model of a photon. Out of the study of NLO we can also model photon absorption and so have a way to describe excited states. (This is discussed in much more detail in my on-line course on TDDFT.) We will not be so ambitious in this course. Our goal in introducing time-dependent perturbation theory is that it is an important step towards understanding the real meaning of the diagrams that we have already seen. However this meaning will not become truly clear until Part II when we will at last have assembled all the tools.
2.1
Adiabatic Perturbation Theory Now let’s not make any sudden moves!
The best application of time-dependent perturbation theory in the context of this course is to rederive time-independent perturbation theory. In order to do this, we carry out so-called adiabatic perturbation theory in which we imagine that our system is initially in its ground stationary state and that the perturbation, Vˆ (t), is turned on adiabatically slowly. Physically this means that timedependent transients have time to die out. Mathematically this means that we have the right to introduce certain convergence factors which help to make integrals converge. We assume that the exact solution is known for the Schr¨odinger equation for the unperturbed system (molecule.) The time-independent equation is, ˆ (0) Ψ(0) = E (0) Ψ(0) . H I I I
(2.1)
The corresponding time-dependent equation is, ˆ (0) Ψ(0) (t) = i ∂ Ψ(0) (t) , H I ∂t I
(2.2)
with, (0)
(0)
(0)
ΨI (t) = ΨI e−iEI 19
t
.
(2.3)
20
CHAPTER 2. TIME-DEPENDENT THEORY (0)
(0)
Careful! The notation is compact: ΨI (t) 6= ΨI . Now apply the time-dependent perturbation, Vˆ (t). The equation governing the time evolution of the perturbed system is, ˆ (0) + Vˆ (t) Ψ0 (t) = i ∂ Ψ0 (t) . (2.4) H ∂t Without loss of generality,
(0)
(0)
(0)
Ψ0 (t) = Ψ0 + δΨ0 (t) + · · · e−iE0
t
,
(2.5)
which allows us to deduce that the linear response of the ground state, δΨ0 (t), satisfies, (0) Vˆ (t)Ψ0
!
∂ ˆ (0) + E0(0) δΨ0 (t) . = i −H ∂t
(2.6)
Note the phase factor in Eq. (2.5). This phase factor is often neglected in elementary treatments of time-dependent perturbation theory (such that found in usual textbook explanation of spectroscopic selection rules.) However the phase factor is indeed important. It is needed to make the perturbed wave function well-defined (theorem of Gell-man and Low) and to make sure that Rayleigh-Schr¨odinger perturbation theory is recovered in the static limit. We will solve this equation using Fourier transformations (Appendix 3.4). First we must rewrite Eq. (2.6) as, Z t h i ˆ (0) − E0(0) δΨ0 (t′ ) dt′ . (2.7) iδΨ0 (t) = Vˆ (t′ )Ψ0 + H −∞
This form is still general. If the perturbation is only applied beginning at some finite time, it is only necessary that Vˆ (t) be defined as zero at all times prior to applying the perturbation. Nevertheless the cases which interest us the most are those for which the perturbation is applied adiabatically (that is, gradually) beginning from time t = −∞ so that transient effects have time to die out and the system response has time to stabilize. The form (2.7) is not yet ideal for carrying out Fourier transforms. The form, iδΨ0 (t) =
Z
+∞
−∞
h
i
ˆ − E0 δΨ0 (t′ ) dt′ , θ(t − t′ ) Vˆ (t′ )Ψ0 + H
(2.8)
is better. Here θ is the Heaviside (or “step”) function, θ(t) =
(
1 ; t>0 0 ; t<0
(2.9)
e+ηt ; t > 0 0 ; t<0
(2.10)
In fact, very soon we will replace θ by θη (t) =
(
which contains a convergence factor with an infinitessimal, η > 0, which guarantees that δΨ(t = −∞) = 0 (i.e., the adiabatic conditon.) We can then apply the Fourier transformation convolution theorem (Appendix 3.4) to obtain, h
i
ˆ − E0 δΨ0 (ω) , iδΨ0 (ω) = θ(ω) Vˆ (ω)Ψ0 + H
(2.11)
21
2.2. PICTURES
but what is the Fourier transformation of the Heaviside function? It is here that we make use of the convergence factor to assure that the Fourier transformation is well-defined: Z
θ(ω) =
+∞
−∞
=
lim+
η→0
=
lim
η→0+
=
lim
η→0+
e+iωt θ(t) dt Z
+∞
−∞ Z +∞
−∞ +∞
Z
0
e+iωt θη (t) dt e+i(ω+iη)t θ(t) dt e+i(ω+iη)t dt
"
ei(ω+iη)t = lim+ η→0 i(ω + iη) +i = lim+ η→0 ω + iη +i . = ω
#+∞ 0
(2.12)
Equation (2.11) becomes, iδΨ0 (ω) = or,
i +i h ˆ ˆ (0) − E0(0) δΨ(0) (ω) , V (ω)Ψ0 + H 0 ω
h
i
(0) ˆ (0) + ω δΨ0 (ω) = Vˆ (ω)Ψ(0) E0 − H 0 .
(2.13)
(2.14)
Equation (2.14) is now in the usual form for applying Rayleigh-Schr¨odinger perturbation theory (Sec. 1.1.) We will assume intermediate normalisation, (0)
hδΨ0 (ω)|Ψ0 i = 0 ,
(2.15)
and then deduce, δΨ0 (ω) =
X
I6=0
(0)
(0)
ΨI
hΨI |Vˆ (ω)|Ψ0i (0)
(0)
ω − (EI − E0 )
.
(2.16)
This is exactly RSPT when the perturbation is static (i.e., ω = 0)! Exercice: Explicitly carry out the intermediate steps between Eq. (2.14) and the solution (2.16). This section contained a number of important lessons, including: (1) the basic idea of adiabatic perturbation theory; (2) the importance of treating phase factors correctly; (3) that time-dependent pertubation theory is formulated in the time representation, (4) but that it is often most conveniently applied in the frequency representation.
2.2
Pictures It is all a question of point of view.
22
CHAPTER 2. TIME-DEPENDENT THEORY
Now that we have seen a bit about how time-dependent theory works, let us focus on the more elegant formulation that is found in most advanced textbooks. This involves the notion of “picture,” and is one of reference frame. We will talk of three pictures, namely the Schr¨odinger, Heisenberg, and interaction pictures, depending upon whether the wave function is varying in time, the operators are varying in time, or both. The most familiar picture is the Schr¨odinger picture. The expectation value of a timeˆ is just, independent operator, A, ˆ ˆ hAi(t) = hΨ(t)|A|Ψ(t)i , (2.17) where the wave function, Ψ(t), satisfies the time-dependent Schr¨odinger equation,
d ˆ (2.18) HΨ(t) = i Ψ(t) . dt Thus operators stay still and wave functions move. In the Heisenberg picture, wave functions stay still and operators move. The wave function in the Heisenberg picture, ˆ ΨH (t) = e+iH(t−t0 ) Ψ(t) , (2.19) where t0 is an arbitrary reference time (often chosen to be zero.) It does not move because, i
d d ˆ ΨH (t) = i e+iH(t−t0 ) Ψ(t) dt dt d ˆ ˆ 0) ˆ +iH(t−t = −He Ψ(t) + ie+iH(t−t0 ) Ψ(t) dt = 0,
(2.20)
so we can just write ΨH instead of ΨH (t). The Heisenberg representation of an operator moves, in the sense that it depends upon time,
It is easy to show that,
ˆ ˆ 0) ˆ −iH(t−t AˆH (t) = e+iH(t−t0 ) Ae .
(2.21)
ˆ hAi(t) = hΨH |AˆH |ΨH i .
(2.22)
ˆ =H ˆ (0) + Vˆ (t) . H
(2.23)
Neither the Schr¨odinger nor the Heisenberg pictures are ideal for doing perturbation theory. As ˆ (0) , and a perturbation, Vˆ (t), usual the Hamiltonian separates into a zero-order part, H
Notice that, although the perturbation is allowed to depend upon time, the zero-order Hamiltonian is static. We will work neither in the Schr¨odinger picture nor in the Heisenberg picture, but rather in the interaction picture defined by, ˆ (0)
Ψint (t) = e+iH (t−t0 ) Ψ(t) ˆ (0) ˆ −iHˆ (0) (t−t0 ) . Aˆint (t) = e+iH (t−t0 ) Ae
(2.24)
It is easily verified that the time-dependent Schr¨odinger equation in the interaction picture is, d Vˆint (t)Ψint (t) = i Ψint (t) . dt Exercise: Verify the correctness of this equation.
(2.25)
In the next section we will define the evolution operator and develop a useful expression based upon the interaction picture.
23
2.3. EVOLUTION OPERATOR
2.3
Evolution Operator Getting there is half the fun.
ˆ t0 ), propagates the wave function in time, The evolution operator, U(t, Ψ(t) = Uˆ (t, t0 )Ψ(t0 ) .
(2.26)
We may also write the same equation in the interaction picture, Ψint (t) = Uˆint (t, t0 )Ψint (t0 ) .
(2.27)
ˆ 0 , t0 ) = ˆ1 . Uˆint (t0 , t0 ) = U(t
(2.28)
d Vˆint (t)Ψint (t) = i Ψint (t) , dt
(2.29)
d ˆ Vˆint (t)Uˆint (t, t0 ) = i U int (t, t0 ) . dt
(2.30)
Of course, Since,
the evolution operator satisfies,
The time-dependence of Vˆint (t) makes the solution of Eq. (2.30) somewhat nontrivial. Direct integration and use of the boundary condition (2.28) gives, Uˆint (t, t0 ) = ˆ1 − i
Z
t
t0
ˆint (t1 , t0 ) dt1 . Vˆint (t1 )U
(2.31)
This equation can be iterated by successive substitution, ˆint (t, t0 ) = ˆ1 − i U = ˆ1 − i =
∞ X
Z
t
t0 Z t t0
dt1 Vˆint (t1 )Uˆint (t1 , t0 ) dt1 Vˆint (t1 ) + (−i)2
Z
t
t0
dt1
Z
t1
t0
ˆint (t2 , t0 ) dt2 Vˆint (t1 )Vˆint (t2 )U
(i) Uˆint (t, t0 )
(2.32)
i=0
where, (0) Uˆint (t, t0 ) = ˆ1 (1) Uˆint (t, t0 ) = −i
Z
t t0
(2) Uˆint (t, t0 ) = (−i)2
dt1 Vˆint (t1 ) Z
(m) Uˆint (t, t0 ) = (−i)m
t
dt1
t0
Z
t
t0
Z
dt1
t1
t0
Z
dt2 Vˆint (t1 )Vˆint (t2 )
t1 t0
dt2 · · ·
Z
tm−1
t0
dtm , Vˆint (t1 )Vˆint (t2 ) · · · Vˆint (tm ) .
(2.33)
The different limits on the nested integrals is somewhat inconvenient for formal manipulations, but can be dealt with by using the time-ordering operator which just commutes the operators so that time always increases from right to left. This is easiest to write down for two operators, h
ˆ 1 )B(t ˆ 2) T A(t h
ˆ 1 )B(t ˆ 2) T A(t
i
i
ˆ 1 )B ˆint (t2 )θ(t1 − t2 ) + B ˆint (t2 )Aˆint (t1 )θ(t2 − t1 ) = A(t =
(
ˆ 1 )B(t ˆ 2 ) ; t1 > t2 A(t ˆ 2 )A(t ˆ 1 ) ; t2 > t1 , B(t
(2.34)
24
CHAPTER 2. TIME-DEPENDENT THEORY
but which generalizes in a straightforward way to any number of operators. Then,
and,
Z t Z t h i (−i)m Z t (m) ˆint dt1 dt2 · · · dtm T Vˆint (t1 )Vˆint (t2 ) · · · Vˆint (tm ) , U (t, t0 ) = m! t0 t0 t0
−i Uˆint (t, t0 ) = T e
Rt
t0
dt′ Vˆint (t′ )
.
(2.35)
(2.36)
It is natural to think that the final perturbed wave function that we want is, ˆ −∞)Ψ0 (−∞) , Ψ(t) = U(t,
(2.37)
but it is not. The one we want is given by the Theorem (Gell-Mann and Low [12])
Ψ(t) =
ˆ −∞)Ψ(0) U(t, 0 (−∞) (0) ˆ −∞)|Ψ(0) hΨ0 (t)|U(t, 0 (−∞)i
(0) Uˆint (t, −∞)Ψ0 , Ψint (t) = (0) ˆ (0) hΨ0 |U int (t, −∞)|Ψ0 i
(2.38)
where we have used the fact that,
ˆ (0) t
(0)
ΨI,int(t) = e+iH
ˆ (0) t
(0)
ΨI (t) = e+iH
(0)
(0)
ΨI e−iEI
t
(0)
= ΨI .
(2.39)
We will not try to prove this theorem. A proof may be found, for example, in Ref. [13] pp. 61-64. More important than the proof is to see for ourselves how the theorem works. Usually, in adiabatic, we are interested Exercise: Let us apply the Gell-Mann and Low theorem to deduce once again the firstorder RSPT wave function. Step 1 Show that, (1)
|Ψint (t)i = =
(0) (0) ˆ (1) ˆ1 − |Ψ(0) 0 ihΨ0 | Uint (t, −∞)|Ψ0 i
X
I6=0
(0)
(0)
(1)
(0)
ˆint (t, −∞)|Ψ0 i . |ΨI ihΨI |U
(2.40)
Notice how the projector, (0) ˆ = ˆ1 − |Ψ(0) Q 0 ihΨ0 | =
X
I6=0
(0)
(0)
|ΨI ihΨI | ,
(2.41)
arises from the denominator in the Gell-Mann and Low theorem. Step 2 We now need to evaluate the matrix elements, (0)
(1)
(0)
ˆint (t, −∞)|Ψ0 i = −i hΨI |U
Z
t −∞
(0)
(0)
hΨI |Vˆint (t1 )|Ψ0 idt1 .
(2.42)
This requires us to do two things. The first is to define the reference time for the interaction picture. We will take this to be the present time (t). The second is to introduce the appropriate form for adiabatic perturbation theory—namely, Vˆ (t1 ) = Vˆ e+η(t1 −t) . Evaluate the integral.
(2.43)
25
2.4. LINKED CLUSTER THEOREM Step 3 Put everything together to get, (1)
|Ψint (t)i =
X
I6=0
(0)
(0)
|ΨI i
(0)
hΨI |Vˆ |Ψ0 i (0)
(0)
E0 − EI + iη
.
(2.44)
At this point we can neglect the infinitessimal η = 0+ . The solution is given in Appendix 3.6.
2.4
Linked Cluster Theorem Come together.
Had we neglected the denominator in the Gell-Mann and Low theorem, then the diagrams for the first-order wave function would have been,
The diagram on the left is unlinked (also called “disconnected”) while that on the right is linked. This is a bit more obvious looking at the corresponding energy diagrams,
A consequence of the Gell-Mann and Low theorem is the Theorem (Linked Cluster) The wave function is the sum of only terms corresponding to linked diagrams. The corresponding energy expression is [9], (0)
(0)
E0 = E0 + hΨ0 |Vˆ
∞ X
ˆ (0) (E0 )Vˆ R
n=0
n
(0)
|Ψ0 iL .
(2.45)
The subscript L means that the corresponding diagrams must be linked. The term, ˆ (0) (ω) = R
(0) (0) (0) X |Ψ(0) 1 |Ψ0 ihΨ0 | I ihΨI | , + (0) ˆ (0) I6=0 ω − E (0) ω−H ω − E0 I
(2.46)
is the (zero-order) resolvant. It is closely related to the (zero-order) Green’s function, ˆ (0) t +iH
ˆ (0) (t) = −ie G ˆ (0) (ω) = G
X
I6=0
(0)
(0)
(0)
(0)
|ΨI ihΨI |θ(t) − |Ψ0 ihΨ0 |θ(−t)
(0) X |Ψ(0) I ihΨI | (0) I6=0 ω − EI + iη
(0)
+
(0)
|Ψ0 ihΨ0 | (0)
ω − E0 − iη
,
(2.47)
26
CHAPTER 2. TIME-DEPENDENT THEORY
which we will see more of later (at which point it will become clear that each lines in the diagrams represents a Green’s function.) Proof A full proof of the linked cluster theorem is quite tedious, but we can give the basic idea diagrammatically. The expansion of the numerator of the wave function in the Gell-Mann and Low theorem gives,
This includes unlinked diagrams where the unlinked parts occur in all possible time orderings. It is this fact which allows us to factor the diagrams to obtain,
Finally we identify the denominator in the Gell-Mann and Low theorem,
27
2.4. LINKED CLUSTER THEOREM
♥ Exercise: Derive expressions for the third-order part of the RSPT wave funcction and show that they can be represented by the diagrams,
Notice that these are just all possible time orderings of a single linked diagram. Note that this involves the trick, 1 1 1 = + . xy x(x + y) y(x + y)
(2.48)
In the the present instance, 1 (0) (E0
−
(0) (0) EI )2 (E0
−
(0) EJ )
= +
1 (0) (E0 (0)
−
(0) (0) EI )(E0
1 (0)
−
(0)
(0) (0) EJ )(2E0 (0)
(0)
(E0 − EI )2 (2E0 − EI − EJ )
The solution is given in Appendix 3.7.
(0)
(0)
− EI − EJ ) .
(2.49)
28
CHAPTER 2. TIME-DEPENDENT THEORY
Chapter 3 Appendices 3.1
Units
This appendix discusses the inevitable question of choice of units. While the official tendancy is to recommend the use of Syst`eme Internationnel (SI) units, the tendancy in electromagnetism is still the use of non SI units, such as Gaussian units. Furthermore theorists overwhelmingly use atomic units (AU), a system based upon Gaussian, rather than SI, units. The objective of this little section is to take stock of these different systems of units. Let us first consider the classiccal expression for the total energy of a molecule. For N electrons and M nuclei, a good approximation for the energy is, in Gaussian units, Etotale = + −
M M X ZI ZJ e2 |pI |2 X + ~ ~ I=1 J=I+1 |RI − RJ | I=1 2mI M X
N X
N X M |pi |2 X e2 + ri − ~rj | i=1 2me i=1 j=1 |~
N X M X
ZI e2 , ~ ri | i=1 I=1 |RI − ~
(3.1)
where me is the mass of the electron and e = |e| is (minus) the charge of an electron. In the Gaussian system of electromagnetic units, the force between two charges is given by Q1 Q2 F~ = rˆ r2 Q1 Q2 = ~r , r3 where
x ~r = y . z
(3.2)
(3.3)
The potential corresponding to this force is given by V =−
Q1 Q2 , r
29
(3.4)
30
CHAPTER 3. APPENDICES
because of the definition ~ F~ = −∇V ∂V /∂x = − ∂V /∂y , ∂V /∂z where
(3.5)
∂/∂x ~ ∇ = ∂/∂y ∂/∂z
(3.6)
is the gradient operator. In the Gaussian system, we use the units centimeter (distance), gram (mass), second (time), and stat coulomb or electrostatic unit (esu, charge). On the other hand, syst`eme internationnel (SI) units are often those officially recommended by journals and for teaching (though the tendancy in some areas of research is not to follow these recommendations.) In SI units, distance is measured in meters, mass in kilograms, time in seconds, and charge in coulombs. It is important to understand the the unit of charge does not have the same dimensionality in Gaussian and SI units. It is thus not too surprising that there is a constant with units which enters into the SI expression for the force between two charges, Q′ Q′ F~ = 1 22 rˆ , 4πǫ0 r
(3.7)
where ǫ0 = 8,854 × 10−12 C2 /N·m2 is the permativity of free space. In comparing with Eqs. (3.2) and (3.7), we see that √ (3.8) Q′ = 4πǫ0 Q , a conversion factor between the two systems of units which is good enough for this course. The potential corresponding to the force given in Eq. (3.7) is in given in SI units, V =−
Q′1 Q′2 . 4πǫ0 r
(3.9)
Finally the system of atomic units (AU) is also very much used in the field of quantum chemistry. The UA system is based dupon the Gaussian system discussed above. It further choses h ¯ , the absolute value of the charge of an electron, and the mass of an electron to be equal to one atomic unit. Alternatively (and more correctly) by using AU’s we have decided to express all quantities in terms of only the 3 basic quantities e, me , and h ¯ . This leads us to the following table: quantity angular momentum charge mass distance energy (linear) momentum
AU h ¯ e me a0 = h ¯ 2 /me e2 Eh = e2 /a0 = me e4 /¯ h2 e2 /¯ h etc.
name
bohr hartree
Note that only some AUs have names, such as bohr for distance and hartree for energy.
3.2. PROOF OF THE 2N + 1 RULE
3.2
31
Proof of the 2n + 1 Rule
Case m = 1. We know that, (n)
EI Since,
= hI (0) |Vˆ |I (n−1) i .
(3.10)
(0) ˆ (0) + hI (0) |E (1) , hI (0) |Vˆ = hI (1) | EI − H I
then,
(3.11)
(0) ˆ (0) |I (n−1) i + hI (0) |I (n−1) iEI(1) = hI (1) | EI − H
(n)
EI
(0) ˆ (0) |I (n−1) i + δn,1 EI(1) . = hI (1) | EI − H
(3.12)
But,
(0) ˆ (0) |I (n−1) i = Vˆ |I (n−2) i − EI − H
n−1 X j=1
(j)
|I (n−1−j) iEI ,
(3.13)
so, (n)
EI
= hI (1) |Vˆ |I (n−2) i −
n−1 X
= hI (1) |Vˆ |I (n−2) i −
n−2 X
j=1
(j)
(1)
(3.14)
(j)
(1)
(3.15)
hI (1) |I (n−1−j) iEI + δn,1 EI ,
or rather, (n)
EI
j=1
hI (1) |I (n−1−j) iEI + δn,1 EI ,
because hI (1) |I (0) i = 0. For n = 3, we obtain the important relation, (3) (2) EI = hI (1) |Vˆ |I (1) i − hI (1) |I (1) iEI .
Case m = 2. Since,
(3.16)
(0) ˆ (0) + hI (1) |E (1) + hI (0) |E (2) , hI (1) |Vˆ = hI (2) | EI − H I I
then, (n) EI
= hI
(1)
= hI
(2)
−
n−2 X j=1
|Vˆ |I (n−2) i − |
(0) EI
n−2 X j=1
(j)
(3.17)
(1)
hI (1) |I (n−1−j) iEI + δn,1 EI
ˆ (0) |I (n−2) i + hI (1) |I (n−2) iEI(1) + hI (0) |I (n−2) iEI(2) −H (j)
(1)
hI (1) |I (n−1−j) iEI + δn,1 EI
(0) ˆ (0) |I (n−2) i − = hI (2) | EI − H
n−3 X j=2
(j)
hI (1) |I (n−1−j) iEI +
2 X
(n)
δn,k EI ,
(3.18)
k=1
But,
(0) ˆ (0) |I (n−2) i = Vˆ |I (n−3) i − EI − H
n−2 X j=1
(j)
|I (n−2−j) iEI ,
(3.19)
32
CHAPTER 3. APPENDICES
so, (n)
EI
= hI (2) |Vˆ |I (n−3) i −
n−2 X
(0) ˆ (0) |I (n−2) i − = hI (2) | EI − H
j=1
(n)
j=1
= hI (2) |Vˆ |I (n−3) i − = hI −
(j)
hI (2) |I (n−2−j) iEI − (j)
hI (2) |I (n−2−j) iEI −
(0)
j=2
(n)
δn,k EI
k=1
n−2 X j=2
2 X
(j)
hI (1) |I (n−1−j)iEI +
(2)
(n)
δn,k EI
k=1
(j)
hI (1) |I (n−1−j)iEI +
(1)
(3)
n−3 X j=1
(0) EI
|
n−3 X j=1
ˆ (0)
−H
(j)
hI (2) |I (n−2−j) iEI −
|I
(n−3)
(j)
hI (2) |I (n−2−j) iEI −
i + hI
n−2 X
2 X
(n)
δn,k EI . (3.20)
k=1
(3)
j=2
3 X
(2)
|I
(n−3)
n−2 X j=2
(j)
hI (1) |I (n−1−j) iEI +
(1) iEI (j)
+ hI
hI (1) |I (n−1−j)iEI +
(0) ˆ (0) |I (n−3) i − = hI (3) | EI − H
+
n−2 X
2 X
ˆ (0) + hI (2) |EI + hI (1) |EI + hI (0) |EI , hI (2) |Vˆ = hI (3) | EI − H
then, EI
j=2
(j)
hI (1) |I (n−1−j) iEI +
n−3 X
= hI (2) |Vˆ |I (n−3) i − Case m = 3. Since,
n−3 X
n−3 X j=2
(1)
2 X
|I
(n−3)
(2) iEI
2 X
(3.21)
(n)
δn,k EI
k=1 (3)
+ hI (0) |I (n−3) iEI
(n)
δn,k EI
k=1 (j)
hI (2) |I (n−2−j) iEI −
n−2 X j=3
(j)
hI (1) |I (n−1−j)iEI
(n)
δn,k EI .
(3.22)
k=1
But
(0)
ˆ (0) |I (n−3) i = Vˆ |I (n−4) i − EI − H
so, (n) EI
= hI +
(3)
3 X
|
(0) EI
ˆ (0) |I (n−3) i − −H
n−3 X j=2
hI
(2)
|I
(n−2−j)
n−3 X j=1
(j) iEI
(j)
|I (n−3−j) iEI ,
−
(3.23)
n−2 X j=3
(j)
hI (1) |I (n−1−j) iEI
(n)
δn,k EI
k=1
= hI (3) |Vˆ |I (n−4) i − +
3 X
n−3 X j=1
(j)
hI (3) |I (n−3−j) iEI −
n−3 X j=2
(j)
hI (2) |I (n−2−j) iEI −
n−2 X j=3
(j)
hI (1) |I (n−1−j)iEI
(n)
δn,k EI
k=1
= hI (3) |Vˆ |I (n−4) i − +
3 X
n−4 X j=1
(j)
hI (3) |I (n−3−j) iEI −
n−3 X j=2
(j)
hI (2) |I (n−2−j) iEI −
n−2 X j=3
(j)
hI (1) |I (n−1−j)iEI
(n)
δn,k EI
k=1
= hI (3) |Vˆ |I (n−4) i −
3 n−k−1 X X
k=1 j=4−k
(j)
hI (k) |I (n−k−j)iEI +
3 X
k=1
(n)
δn,k EI .
(3.24)
33
3.3. SECOND-ORDER RSPT WAVE FUNCTION Case Any m. By now it is relatively straightforward to obtain the general formula by induction, (n) EI
3.3
= hI
(m)
|Vˆ |I (n−m−1) i −
m X
n−k−1 X
hI
(k)
k=1 j=m−k+1
|I
(n−k−j)
(j) iEI
+
m X
(n)
δn,k EI .
(3.25)
k=1
Second-Order RSPT Wave Function One, two, three, ...
We wish to derive the formula for the second-order wave function, (2)
Ψ0 =
X
(0)
(2)
ΨI CI,0 ,
(3.26)
I=1,∞ (2)
which is to say that we want to derive a formula for the coefficients C1,0 . Notice that intermediate normalization excludes I = 0 from the sum. Since, (0) (2) (1) (1) (2) (0) ˆ (0) Ψ(2) ˆ (1) H 0 + V Ψ0 = E0 Ψ0 + E0 Ψ0 + E0 Ψ0 ,
(3.27)
then,
ˆ (0) − E0(0) Ψ(2) H = 0 =
(1) (1) (2) (0) E0 − Vˆ Ψ0 + E0 Ψ0
X
J=1,∞
(0) ˆ (0) |V |0 i (2) (0) (0) hJ + E0 Ψ0 . h0(0) |Vˆ |0(0) i − Vˆ ΨJ (0) (0) E0 − EJ
(3.28)
(0)
Multigration by ΨI with I 6= 0 gives,
(0)
(0)
EI − E0
(2)
CI,0 =
That is, (2)
CI,0 = − or
h0(0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0)
(0)
E0 − EI
h0(0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0)
(0)
(E0 − EI )2
+
(2)
−
X
hI (0) |Vˆ |J (0) ihJ (0) |Vˆ |0(0) i (0)
hI (0) |Vˆ |J (0) ihJ (0) |Vˆ |0(0) i
X
(0)
(0)
(0)
(a)
Ψ0 = − is represented by diagram (a),
X
(a)
I=1,∞
(0) h0
ΨI
(0)
(0)
J=1,∞ (E0 − EI )(E0 − EJ ) (b)
Ψ 0 = Ψ0 + Ψ0 , where,
(0)
E0 − EJ
J=1,∞
(3.29)
(3.30) (3.31)
|Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0)
,
.
(0)
(E0 − EI )2
,
(3.32)
34
CHAPTER 3. APPENDICES
and, X
(b)
Ψ0 =
hI (0) |Vˆ |J (0) ihJ (0) |Vˆ |0(0) i
(0)
ΨI
(0)
(0)
(0)
(3.33)
(0)
(E0 − EI )(E0 − EJ )
I,J=1,∞
is represented by diagram (b). Notice that the sign associated with diagram (a) is negative because there are two hole lines and one loop while the sign associated with diagram (b) is positive because there is only one hole line and one loop. The corresponding third order energy expressions are, (3)
(a)
(b)
E0 = E0 + E0 ,
(3.34)
with, (a)
E0 = −
X
(0) h0
(0)
ΨI
|Vˆ |I (0) ih0(0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0)
(0)
(E0 − EI )2
I=1,∞
,
(3.35)
and, (b)
E0 =
X
h0(0) |Vˆ |I (0) ihI (0) |Vˆ |J (0) ihJ (0) |Vˆ |0(0) i (0)
(0)
(0)
(0)
(E0 − EI )(E0 − EJ )
I,J=1,∞
.
(3.36)
The energy diagrams are,
3.4
Fourier Transforms Dynamics or spectra?
According to the Fourier transform theorem (for continuous Fourier transforms), Z
f˜(ν) =
+∞
−∞ Z +∞
f (t) =
−∞
e+2πiνt f (t) dt e−2πiνt f˜(ν) dν .
(3.37)
It follows that, f (t) = =
Z
+∞
−∞ Z +∞ −∞
−2πiνt
e
Z
Z
+∞
−∞
+∞
−∞
+2πiνt′
e ′
′
′
f (t ) dt
e+2πiν(t −t) dν f (t′ ) dt′ ,
dν (3.38)
35
3.4. FOURIER TRANSFORMS and thus that,
Z
+∞
−∞
In particular,
e+2πνi(t−t0 ) dν = δ(t − t0 ) .
Z
+∞
−∞
(3.39)
e+2πiνt dν = δ(t) .
(3.40)
This last relation is a powerful mneumonique for the Fourier transform theorem, but cannot be considered as rigorous because of the problems associated with integrating unity from minus infinity to plus infinity when t = 0 and of integrating an oscillating function over the same range when t 6= 0. On the other hand, we can make the same relation rigorous by introducing convergence parameters and taking the limit that the parameters become infinitely small, Z
+∞
−∞
e+2πiνt dν =
lim+
η→0
=
lim
η→0+
Z
0
−∞
"
e+2πiν(t−iη) dν + lim+ η→0
e+2πiν(t−iη) 2πi(t − iη) "
#0
−∞
+ lim+ η→0
"
#
Z
+∞
0
e+2πiν(t+iη) dν
e+2πiν(t+iη) 2πi(t + iη)
#+∞ 0
1 1 i − η→0 2π t − iη t + iη i (t + iη) − (t − iη) lim+ − η→0 2π t2 + η 2 +2iη i lim+ − 2 η→0 2π t + η 2 η lim+ 2 η→0 π(t + η 2 ) δ(t) , lim+ −
= = = = =
(3.41)
where we have used that fact that the δ function can be obtained as the limit of a an infinitely narrow Lorenztian line shape (Sec. 3.5). While the convergence parameter trick may shock a mathematician, it is very useful in physics. We will often use it in adiabatic perturbation theory to eliminate transient response. A second theorem concerns the convolution of two functions, (f ◦ g)(t) =
Z
+∞
−∞
f (t − t′ )g(t′ ) dt′ .
(3.42)
Its Fourier transform is, (fg ◦ g)(ν) = =
= =
Z
+∞
−∞ +∞
e+2πiνt
Z
Z
−∞ Z +∞
−∞ Z +∞
−∞ +∞
Z
−∞
+∞
−∞ +∞
Z
−∞
= f˜(ν)˜ g (ν) .
Z
Z
+∞
e−2πiν (t−t ) f˜(ν ′ ) dν ′ ′
−∞ +∞ +2πi(ν−ν ′ )t
−∞
e
′
Z
dt
+∞
−∞
Z
+∞
−∞
′
e−2πiν”t g˜(ν”) dν” dt′ dt
′ ′ e+2πi(ν −ν”)t dt′ f˜(ν ′ )˜ g (ν”) dν ′ dν”
δ(ν − ν ′ )δ(ν ′ − ν”)f˜(ν ′ )˜ g (ν”) dν ′ dν” δ(ν − ν ′ )δ(ν − ν”)f˜(ν ′ )˜ g (ν”) dν ′ dν” (3.43)
36
CHAPTER 3. APPENDICES
In physics, we are often sloppy and simply drop the tilda (˜). The Fourier transform is then understood through the variable of the function, Z
f (ν) =
+∞
−∞ +∞
Z
f (t) =
−∞
e+2πiνt f (t) dt e−2πiνt f (ν) dν .
(3.44)
Also in physics, the Fourier transforms are not always formulated in a symmetric manner. Thus instead of using the t ↔ ν transform, we can rewrite the same theorems with the t ↔ ω transform, Z
f (ω) =
+∞
−∞
e+iωt f (t) dt
Z
1 +∞ −iωt f (t) = e f (ω) dω 2π −∞ 1 Z +∞ +iω(t−t′ ) e dω δ(t − t′ ) = 2π −∞ Z
(f ◦ g)(t) =
+∞
−∞
f (t − t′ )g(t′ ) dt′
(f ◦ g)(ω) = f (ω)g(ω) , or with the t ↔ E transform,
Z
f (E) =
+∞
−∞
(3.45)
e+iEt/¯h f (t) dt
Z
1 +∞ −iEt/¯h e f (E) dE h −∞ Z 1 +∞ +iE(t−t′ )/¯h ′ δ(t − t ) = e dE h −∞ f (t) =
Z
(f ◦ g)(t) =
+∞
−∞
f (t − t′ )g(t′ ) dt′
(f ◦ g)(E) = f (E)g(E) ,
(3.46)
We will most often use the t ↔ ω transform.
3.5
The Dirac δ Function What is the value of the function right here?
The Dirac δ function is defined in terms of the integral of its product wtih an arbitrary function, f , as, Z ∞ δ(x − x0 )f (x) dx . (3.47) f (x0 ) = −∞
It follows that, f (0) = Doing this requires that, δ(x) =
Z
(
∞
−∞
δ(x)f (x) dx .
0 ; x 6= 0 , +∞ ; x = 0
(3.48)
(3.49)
37
3.6. FIRST-ORDER RSPT WAVE FUNCTION and,
Z
∞ −∞
δ(x) dx = 1 .
(3.50)
If we insist on a rigorous terminology, δ(x) is a distribution rather than a function because of its divergence at x = 0. The δ function may be obtained as the limits of different functions when a small real positive number, η, goes to zero (η → 0+ ). Particularly useful representations include: a Gaussian, 1 2 2 δ(x) = lim+ √ e−x /η ; η→0 η π
(3.51)
η ; + η2)
(3.52)
a Lorentzian, δ(x) = lim+ η→0
π(x2
and the oscillating function, δ(x) = lim+ η→0
sin(x/η) . πx
(3.53)
The δ function has several interesting properties (see for example Ref. [14] p. 57) that follow from its defining integral. For example, the δ function is even, f (0) = = = =
Z
+∞
−∞ Z +∞ −∞ −∞
Z
+∞ Z +∞ −∞
δ(x)f (x) dx δ(x)f (−x) dx δ(−x)f (x) (−dx) δ(−x)f (x) dx ,
(3.54)
hence δ(−x) = δ(x) .
(3.55)
Other relations include,
Z
+∞
−∞
3.6
δ ′ (x) = −δ ′ (x) xδ(x) = 0 xδ ′ (x) = −δ(x) 1 δ(ax) = δ(x) a > 0 a 1 δ(x2 − a2 ) = [δ(x − a) + δ(x + a)] a > 0 2a δ(a − x)δ(x − b) dx = δ(a − b) f (x)δ(x − a) = f (a)δ(x − a) .
First-Order RSPT Wave Function Clear the smoke! Break the mirrors!
(3.56)
38
CHAPTER 3. APPENDICES We apply the Gell-Mann and Low theorem to deduce the first-order RSPT wave function.
Step 1 (0) Uˆint (t, −∞)|Ψ0 i |Ψint (t)i = (0) ˆ (0) hΨ0 |U int (t, −∞)|Ψ0 i
ˆ (1) (t, −∞) + O(2) |Ψ(0) ˆ1 + U 0 i = (0) (1) (0) 1 + hΨ0 |Uˆint (t, −∞)|Ψ0 i + O(2)
=
(0)
(1)
(0)
ˆint (t, −∞)|Ψ0 i + O(2) 1 − hΨ0 |U
ˆ1 + Uˆ (1) (t, −∞) + O(2) |Ψ(0) 0 i
(0) (1) (0) (0) (0) ˆ (1) (0) = |Ψ0 i + Uˆint (t, −∞)|Ψ0 i − |Ψ0 ihΨ0 |U int (t, −∞)|Ψ0 i + O(2) , (3.57)
So,
(1) (0) (0) ˆ (1) (0) ˆ (1) (t, −∞)|Ψ(0) |Ψint (t)i = U 0 i − |Ψ0 ihΨ0 |Uint (t, −∞)|Ψ0 i int (0) (0) (1) (0) = ˆ1 − |Ψ0 ihΨ0 | Uˆint (t, −∞)|Ψ0 i
=
X
I6=0
(0)
(0)
(1)
(0)
ˆint (t, −∞)|Ψ0 i , |ΨI ihΨI |U
(3.58)
where use has been made of the complete orthonormal nature of the zero-order wave functions. Step 2 (0)
(1)
(0)
ˆint (t, −∞)|Ψ0 i = −i hΨI |U = −i = −i
Z
t
(0)
−∞ Z t
−∞ t
Z
−∞
(0)
hΨI |Vˆint (t1 )|Ψ0 idt1 ˆ (0) (t1 −t) ˆ +η(t1 −t) −iH ˆ (0) (t1 −t)
(0)
hΨI |e+iH
(0)
(0)
hΨI |e+iEI
(0)
(0)
= −ihΨI |Vˆ |Ψ0 i (0)
(0)
= −ihΨI |Vˆ |Ψ0 i = =
Z
(0) (0) −ihΨI |Vˆ |Ψ0 i (0)
hΨI |Vˆ |Ψ0 i (0)
Ve
t
−∞
e
(0)
(t1 −t) ˆ +η(t1 −t) −iE0 (t1 −t)
−∞ Z 0
(0) (0) −ihΨI |Vˆ |Ψ0 i
(0)
=
Ve
e
(I)
−E0 −iη)(t1 −t)
(I)
−E0 −iη)t′
e+i(EI
e+i(EI (I)
e+i(EI
(0)
(0)
(0)
−E0 −iη)t′
(I)
(0)
i(EI − E0 − iη) 1 (I)
(0)
|Ψ0 idt1 (0)
|Ψ0 idt1
dt1
dt′
0
−∞
(0)
i(EI − E0 − iη) (3.59)
(I)
E0 − EI + η
Step 3 (1)
|Ψint (t)i = =
X
|ΨI ihΨI |Uˆint (t, −∞)|Ψ0 i
X
|ΨI i
I6=0
I6=0
(0)
(0)
(1)
(0)
(0)
(0)
(0)
hΨI |Vˆ |Ψ0 i (0)
(0)
E0 − EI + η
.
(3.60)
39
3.7. THIRD-ORDER RSPT WAVE FUNCTION
3.7
Third-Order RSPT Wave Function The answer is simpler than the demonstration.
We want to show that the third-order part of the RSPT wave function may be expressed by the diagrams,
From Sec. 1.1, (1) (2) (2) (1) (3) (0) ˆ (0) Ψ(3) ˆ (2) = E0(0) Ψ(3) H 0 + V Ψ0 0 + E0 Ψ0 + E0 Ψ0 + E0 Ψ0 (0)
(3)
(1)
(2)
(2)
(1)
= E0 Ψ0 + E0 Ψ0 + E0 Ψ0 ,
(3.61)
where the last term disappears due to the intermediate normalization criterion. So, (0)
(3)
(1)
(2)
(2)
(1)
ˆ (0) )Ψ0 = (Vˆ − E0 )Ψ0 − E0 Ψ0 . (E0 − H
(3.62)
Then, (3) Ψ0
=
X
(0) hI ΨI
I6=0
From Appendix 3.3, we have that (2)
Ψ0 = −
X
(0) h0
(0)
ΨJ
J=1,∞
(0)
(1)
|Vˆ − E0 |0(2) i (0)
(0)
E0 − EI
|Vˆ |0(0) ihJ (0) |Vˆ |0(0) i (0)
(0)
(E0 − EJ )2
+
−
X
(2) (0) (1) |0 i (0) E0 hI ΨI (0) (0) E0 − EI I6=0
X
(0) hJ
ΨJ
J,K=1,∞
(0)
.
(3.63)
|Vˆ |K (0) ihK (0) |Vˆ |0(0) i
(0)
(0)
(0)
(0)
(E0 − EJ )(E0 − EK )
.
(3.64)
(3)
Hence the first part of Ψ0 is, X
I6=0
(0) hI
ΨI
(0)
|Vˆ |0(2) i
(0)
(0)
E0 − EI
= − +
X
(0) hI
ΨI
I,J6=0
X
|Vˆ |J (0) ih0(0) |Vˆ |0(0) ihJ (0) |Vˆ |0(0) i (0)
(0)
(0)
(0)
(E0 − EI )(E0 − EJ )2 (0) ˆ |V |J (0) ihJ (0) |Vˆ |K (0) ihK (0) |Vˆ |0(0) i (0) hI
ΨI
I,J,K6=0 (a)
(0)
(c)
(0)
(0)
(0)
(0)
(0)
(0)
(E0 − EI )(E0 − EJ )(E0 − EK )
= Ψ0 + Ψ0 ,
where the superscript letter indicates the corresponding diagram.
(3.65)
40
CHAPTER 3. APPENDICES Since, (1) E0 = h0(0) |Vˆ |0(0) i ,
(3.66)
(3)
then the second part of Ψ0 is, (1) (0) (2) |0 i (0) E0 hI − ΨI (0) (0) E0 − EI I6=0
X
X
=
(0) h0
ΨI
I6=0
X
−
(0)
|Vˆ |0(0) i2 hI (0) |Vˆ |0(0) i (0)
(0)
(E0 − EI )3 (0) ˆ (0) (0) ˆ |V |J (0) ihJ (0) |Vˆ |0(0) i (0) h0 |V |0 ihI
ΨI
(0)
I,J6=0 (d)
(b)
= Ψ0 + Ψ0 ,
(0)
(0)
(0)
(E0 − EI )2 (E0 − EJ )
(3.67)
From Appendix 3.6, we have that, (1)
Ψ0
X
=
(0) hI
(0)
ΨI
(2)
(0)
(0)
E0 − EI ˆ |J (0) ihJ (0) |Vˆ |0(0) i X h0(0) |V
I6=0
E0
|Vˆ |0(0) i
=
(0)
.
(0)
E0 − EJ
J6=0
(3.68)
(3)
It follows that the third part of Ψ0 is, (2) (0) (1) |0 i (0) E0 hI − ΨI (0) (0) E0 − EI I6=0
X
= − = − −
X
ΨI
X
ΨI
(0) h0
I,J6=0 (e)
(0)
(0)
I,J6=0 (0)
ΨI
|Vˆ |J (0) ihJ (0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0)
(0)
(0)
(E0 − EJ )(E0 − EI )2 (0) ˆ (0) ihJ (0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i (0) h0 |V |J
I,J6=0
X
(0)
(0)
(0)
(0)
(0)
(E0 − EI )2 (2E0 − EI − EJ ) h0(0) |Vˆ |J (0) ihJ (0) |Vˆ |0(0) ihI (0) |Vˆ |0(0) i
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(E0 − EI )(E0 − EJ )(2E0 − EI − EJ )
(f )
= Ψ0 + Ψ0 ,
(3.69)
where use has been made of Eq. (2.49).
3.8
Rules for One-Particle Energy Diagrams
Summary of rules for single-particle energy diagrams (Ref. [15] pp. 327-349). These are simpler than diagrams with electron repulsions because there are no factors of 1/2 about which to worry.
• Rule 0 Draw all topologically distinct linked diagrams.
3.8. RULES FOR ONE-PARTICLE ENERGY DIAGRAMS
41
ˆ ˆ • Rule 1 hin|h|outi = hr|h|si.
• Rule 2 Each horizontal cut,
contributes a denominator of, X
hole
ǫi −
X
particle
ǫa = (ǫi + ǫj + · · ·) − (ǫa + ǫb + · · ·) .
(3.70)
• Rule 3 There is a prefactor of (−1)h+l where h is the number of hole lines and l is the number of loops. Note that always l = 1 for 1 particle. • Rule 4 Sum over all particle and hole indices.
42
CHAPTER 3. APPENDICES
Part II Many-Body Perturbation Theory
43
45 Two Elegant Formalisms Dirac: |ψi1 ψi2 · · · ψiN | → |i1 i2 . . . iN i
(3.71)
Second Quantization: ˆ H ˆ H
= → =
X ˆ +1 vˆ(i, j) h(i) 2 i,j=1,N i=1,N X
X ij
hi,j a ˆ†i aˆj +
1X ˆl aˆk ˆ†j a vij,kl a ˆ†i a 4 ijkl
(3.72)
And the one was many. Up to this point we have completely neglected the nature of our system. What distinguishes many-body perturbation theory (MBPT) is that we take explicitly into account that our system is composed of N indistinguishable particles. Although MBPT applies to boson systems as well as fermion systems, we shall restrict ourselves to electrons which are fermions. This allows various interesting properties to be proven such as size extensivity (the energy per particle is independent of the size of the system for a large homogenous system) and size consistency (the total energy of a supersystem consisting of two subsystems separated by a large distance is just the sum of the energies of the two subsystems.) While these are interesting properties which follow automatically in properly formulated MBPT, we will focus on more elementary considerations such as integral evaluation and the meaning of the diagrams. Our principle emphasis will be on second quantization. This is is a formalism which emphasizes operators and 1- and 2-electron matrix elements rather than Slater determinants. Sometimes it is said that any manipulation done with second quantization could also be done without this formalism. Thus the only reasons to use second quantization are its elegance and its frequent use in the literature. However this reasoning is not quite correct because the fact that N never appears in the fundamental ˆ also means that systems with an unknown or variable number of electrons may be equation for H treated. Such problems frequently arise for solids. For us, second quantization is in the first instance “just” an algebraic machine that we can use in developing computationally useful equations. In the time-independent theory, second quantization is simply a way to evaluate matrix elements which occur in RSPT. Indeed many of the results which are obtainable with sophisticated Green’s function MBPT may also be obtained using wave function techniques such as RSPT [1, 3]. The initial description of second quantization employed here is very mathematical. However I have reduced the size of the type for those details which I consider to be less important. In the second instance, second quantization and time-dependent perturbation theory provides direct insight into diagrams. This is because Wick’s theorem leads directly to Green’s functions which are represented as lines in our diagrams, which brings us to the end of our course.
46
Chapter 4 Time-Independent Theory 4.1
Creation and Annihilation Operators Add one. Take away two.
The language of second quantization supposes a predetermined underlying one-electron (spin) orbital basis set, {ψ1 , ψ2 , ψ3 , . . .}. This basis set should be complete and orthonormal. hi|ji =
Z
ψi∗ ψj dτ = δi,j .
(4.1)
It allows us to construct kets which correspond to Slater determinants: |i |ii |iji |ijki .. . etc.
(4.2)
The state |i is special. It is the Slater determinant corresponding to zero electrons and is called the vacuum state. Definition. HN is the (Hilbert) space of N-electron functions. We thus have |i |ii |iji |ijki
∈ ∈ ∈ ∈ .. . etc.
H0 H1 H2 H3 (4.3)
Definition. The creation operator is defined by a ˆ†l |ij . . .i = |lij . . .i . 47
(4.4)
48
CHAPTER 4. TIME-INDEPENDENT THEORY We thus have that a ˆ†l : HN → HN +1 .
(4.5)
In fact we see that Slater determinants are not among the most fundamental building blocks of the formalism because we can always build them by applying creation operators to the vacuum state. |i a ˆ†i |i = |ii ˆ†i |ji = |iji ˆ†j |i = a a ˆ†i a
a ˆ†i a ˆ†j aˆ†k |i = a ˆ†i a ˆ†j |ki = a ˆ†i |jki = |ijki ···
(4.6)
Let us consider in more detail the properties of creation operators. Definition The anticommutator ˆ B} ˆ = AˆB ˆ +B ˆ Aˆ . {A,
(4.7)
{ˆ a†i , a ˆ†j } = 0 .
(4.8)
Theorem Proof An operator is defined by its action. {ˆ a†i , a ˆ†j }|kl . . .i
=
a ˆ†i a ˆ†j + a ˆ†j a ˆ†i |kl . . .i
= |ijkl . . .i + |jikl . . .i
= |ijkl . . .i − |ijkl . . .i = 0 ♥
(4.9)
Definition The adjoint, aˆi , of the creation operator, a ˆ†i , is the corresponding annihilation operator. Theorem {ˆ ai , a ˆj } = 0 . (4.10) Proof 0 = =
♥
{ˆ ai , a ˆ j }† (ˆ ai a ˆj )† + (ˆ aj a ˆi )†
=
a ˆ†j a ˆ†i + a ˆ†i a ˆ†j
=
ˆ†j } {ˆ a†i , a
(4.11)
Let us take stock of the action of a ˆi . Let us look at a simple example, a ˆ†i a ˆi
: HN → HN +1
: HN → HN −1 .
We have used the closure relation, ˆ 1=
X
|jihj| .
(4.12)
(4.13)
Notice my tendancy to use a pseudo-Einstein convention: In the absence of explicit indices on a sum sign, then it is understood that the sum is over repeated indices unless otherwise specified.
4.1. CREATION AND ANNIHILATION OPERATORS
49
In order to continue, we need the identity, Z Z 1 ∗ ∗ hmn|iji = (ψm (1)ψn∗ (2) − ψn∗ (1)ψm (2))(ψi (1)ψj (2) − ψj (1)ψi (2)) d1d2 2 Z Z ∗ = ψm (1)ψn∗ (2)(ψi (1)ψj (2) − ψj (1)ψi (2)) d1d2 Z Z Z Z ∗ ∗ = ψm (1)ψi (1) d1 ψn∗ (2)ψj (2) d2 − ψm (1)ψj (1) d1 ψn∗ (2)ψi (2) d2 =
=
hm|iihn|ji − hm|jihn|ii δm,i δn,j − δm,j δn,i .
(4.14)
Then a ˆi |mni =
X
|jihj|ˆ ai |mni
X
|jihmn|iji∗
X
= =
X
=
X
=
|jihmn|ˆ a†i |ji∗ |jihij|mni |ji (δm,i δn,j − δm, jδn,i )
= δm,i |ni − δn,i |mi |ni ; i = m −|mi ; i = n = 0 ; autrement
In general we have the Theorem
a ˆi |kl . . .i = 0 si i ∈ /{k, l, . . .} ,
aˆi |kl . . . i . . .i = (−1)t |kl . . . /i . . .i , where t is the number of transpositions needed to create the determinant |ikl . . .i. Corollary aˆ†i a ˆi |kl . . .i = ni |kl . . .i , where ni =
(
0; i ∈ /{k, l, . . .} . 1 ; i ∈ {k, l, . . .}
(4.15)
(4.16) (4.17)
(4.18) (4.19)
Definition n ˆi = a ˆ†i aˆi
(4.20)
{ˆ ai , a ˆ†j } = δi,j
(4.21)
{ˆ ai , a ˆ†j }|kl . . .i = a ˆi a ˆ†j + a ˆ†j a ˆi |kl . . .i
(4.22)
δi,j |kl . . .i .
(4.23)
is the number operator for orbital i. Theorem Proof Consider
There are two cases, depending upon whether i is in {kl . . .} or not. Case 1 : i ∈ /{kl . . .}. Then ˆi |jkl . . .i a ˆi a ˆ†j + a ˆ†j a ˆi |kl . . .i = a =
50
CHAPTER 4. TIME-INDEPENDENT THEORY
case 2 : i ∈ {kl . . .}. Then ˆi |jkl . . . i . . .i + (−1)t a ˆ†j |jkl . . . i/ . . .i a ˆi a ˆ†j + a ˆ†j a ˆi |kl . . . i . . .i = a
= (1 − δi,j )ˆ ai |jkl . . . i . . .i + (−1)t a ˆ†j |kl . . . i/ . . .i
= (1 − δi,j )(−1)t+1 |jkl . . . i/ . . .i + (−1)t |jkl . . . i/ . . .i = δi,j (−1)t |jkl . . . i/ . . .i = δi,j |kl . . . i . . .i .
♥
(4.24)
Lazy Notation i = a ˆi i = a ˆ†i . †
(4.25)
This notation is handy for long algebraic manipulations, but may lead to confusion. Nevertheless this lazy notation is more and more used in the literature. Our anticommutation relations may be summarized in this notation as {i, j} = {i† , j † } = 0 {i, j † } = δi,j .
4.2
(4.26)
Second-Quantized Operators Where is N?
We can make use of the work done up to now to write the Condon-Slater rules in the form, Theorem X
1X hij||klii† j † lk 4 X 1X † ˆ (ik||jl)i† j † lk . = hi|h|jii j+ 2
ˆ = H
† ˆ hi|h|jii j+
(4.27)
Reminder hij|ˆ v|kli = hij||kli = (ik||jl) − (il||jk) (Notation de Mullikan) .
(4.28)
More concretely, (ij||kl) = (ij|fH |kl) =
Z Z
ψi∗ (x1 )ψj (x1 )
1 ∗ ψ (x2 )ψl (x2 ) dx1 dx2 , r12 k
(4.29)
where, fH (x1 , x2 ) = fHσ,τ (r1 , r2 ) 1 , = r12
(4.30)
51
4.3. DENSITY MATRICES is the Hartree kernel. Thus X
hij||klii† j † lk =
X
=
X
X
=
= 2
(ik||jl)i† j † lk − (ik||jl)i† j † lk +
(ik||jl)i† j † lk +
X
(ik||jl)i† j † lk .
X
X
X
(il||jk)i† j † lk
(il||jk)i† j † kl (ik||jl)i† j † lk (4.31)
Proof of the theorem It suffices to show that we obtain the same expressions for the matrix elements in the basis of Slater determinants that we have already obtained explicitly. For example: Given |Ii = |i1 i2 . . . iN i . (4.32)
Then
ˆ hI|h|Ii = hI|
X
= Evidently except if i, j ∈ {i1 , i2 , . . . , ıN }. Now So ˆ hI|h|Ii = =
X
† ˆ hi|h|jii j |Ii
† ˆ hi|h|jihI|i j|Ii .
(4.33)
hI|i† j|Ii = 0
(4.34)
i† j = δi,j − ji† .
(4.35)
X
i,j∈I
X i∈I
ˆ δi,j hI|Ii − hI|ji† |Ii hi|h|ji
ˆ , hi|h|ii
| {z } 1
|
{z 0
}
(4.36)
which is the same result given by the Slater-Condon rules. Etc. etc. ♥
4.3
Density Matrices Reduced-body theory
Corollary Given Aˆ(1) =
X
a ˆ(1) (i)
i=1,N (i
Aˆ(2) =
X
a ˆ(2) (ij) .
(4.37)
i,j=1,N
Then the expectation values for the state |Ψi are given by X hAˆ(1) i = hi|ˆ a(1) |jiγj,i
tr a(1) γ 1X hij|ˆ a(2) |kliΓkl,ij hAˆ(2) i = 4 =
(i
= =
X
hij|ˆ a(2) |kliΓkl,ij
tr a(2) Γ ,
(4.38)
52
CHAPTER 4. TIME-INDEPENDENT THEORY
where γj,i = hΨ|i† j|Ψi
(4.39)
Γkl,ij = hΨ|i† j † lk|Ψi
(4.40)
tr γ = N tr Γ = N(N − 1) .
(4.41)
is the 1-electron density matrix and
is the 2-electron density matrix. Theorem
Proof tr γ
=
=
X
= =
hˆ ni N,
=
where n ˆ= is the number operator. tr Γ
= = = = = =
X
X
X
X
γi,i hi† ii hˆ ni i
n ˆi
(4.42) (4.43)
Γij,ij
X
hi† j † jii X − hj † i† jii X X − δi,j hj † ii + hj † ji† ii X X hj † ji† ii − hi† ii X X hˆ nj n ˆii − hˆ ni i
= hˆ n (ˆ n − 1)i
= N (N − 1) .
(4.44)
♥
4.4
Particle-Hole Representation Keep your eye on the donut not on the hole.
Consider a “change of coordinate” defined with respect to a Slater determinant, |Φi = |123 . . . N i . . . .
(4.45)
53
4.4. PARTICLE-HOLE REPRESENTATION c b a i j k
----------||-||-||-
N, N-1 N-2, N-3 N-4, N-5
. . . We will introduce a new Index Convention | ijklmn | opq . . . xyz abc . . . gh | {z } | {z } | {z } virtual orbitals occupied orbitals free orbitals
The operators
(4.46)
cˆ†i = a ˆi
(4.47)
cˆ†a = a ˆ†a
(4.48)
“create holes” and the operators “create particles” in the “vacuum state” |Φi. . . . c b a i j k
-----x----||-|O-||-
particule
trou
. . . As {ˆ cr , cˆs }
{ˆ cr , cˆ†s }
= {ˆ c†r , cˆ†s } = 0
= δi,j ,
(4.49)
we recover all the algebra of second quantization except that the form of the hamiltonian becomes more complicated. Definition A second quantized operator is said to be normally ordered when all the annihilation operators are to the right of all the creation operators. Theorem X X X X ˆ − hHi ˆ H = Fa,b cˆ†a cˆb − Fi,j cˆ†j cˆi + Fi,b cˆi cˆb + Fa,j cˆ†a cˆ†i 1X 1X = (ab||cd)ˆ c†a cˆ†c cˆd cˆb + (ij||kl)ˆ c†l cˆ†j cˆi cˆk 2 2 X X + (ab||kl)ˆ c†a cˆ†k cˆb cˆl − (ib||cl)ˆ c†a cˆ†j cˆd cˆk X X + (aj||kl)ˆ c†a cˆ†l cˆ†j cˆk − (aj||cd)ˆ c†a cˆ†c cˆ†j cˆd X X + (ab||kd)ˆ c†a cˆd cˆb cˆk − (ij||kd)ˆ c†j cˆd cˆi cˆk 1X 1X (ib||kd)ˆ cd cˆb cˆi cˆk + (aj||cl)ˆ c†a cˆ†c cˆ†l cˆ†j , (4.50) + 2 2
54
CHAPTER 4. TIME-INDEPENDENT THEORY
where
ˆ + Fr,s = hr|h|si
X
(ii||rs) −
X
(ir||is)
(4.51)
is the matrix of the Fock operator. Proof A lot of algebra. ♥ For us, the particle-hole representation is more a source of terminology than a practical aid for algebraic manipulations.
4.5
Wick’s Theorem (simplified version)
Wick’s theorem refers to a previously determined determinant of Slater (the physical vacuum state),, |Φi = |123 . . . Ni .
(4.52)
We will also use the Orbital Index Convention abc . . . gh | ijklmn | opq . . . xyz |
{z
}
| {z }
|
{z
(4.53)
}
unoccupied occupied unspecified but not the particle-hole operators, cˆr and cˆ†s , of the particle-hole representation. Definition The contraction of two second-quantized operators (that is, of two creation and annihilation operators or of two creation operators or of two annihilation operators) are the numbers given by r (1) s(1) r †(1) s†(1) r (1) s†(1) r †(1) s(1) To evaluate an expression like
= = = =
hΦ|rs|Φi = 0 hΦ|r † s† |Φi = 0 hΦ|rs† |Φi = δr,s (1 − ns ) = δr,s n ¯s † hΦ|r s|Φi = δr,s ns .
(4.54)
(1) (2) (1) (2) (3) (3)
hΦ|b1 b2 b3 b4 b5 b6 |Φi
(4.55)
where the bi are creation and annihilation operators, it suffices to apply the following rules: (k) (k)
1. Anticommute the bi without changing their order within each pair bi bj . In our example we get (1) (1) (2) (2) (3) (3) −hΦ|b1 b3 b2 b4 b5 b6 |Φi . (4.56) Handy Trick : If we write
(1) (2) (1) (2) (3) (3)
hΦ|b1 b2 b3 b4 b5 b6 |Φi
and connect each contracted pair with a line,
+-----+ +--|--+ | +--+ | | | | | | < Phi | b b b b b b | Phi > 1 2 3 4 5 6
(4.57)
55
4.6. HARTREE-FOCK ENERGY EXPRESSION
the sign after reordering is determined by whether the number of intersections of lines is odd or even. 2. Calculate the product
(1) (1) (2) (2) (3) (3)
−b1 b3 b2 b4 b5 b6
(4.58)
hΦ|ˆb1ˆb2 . . . ˆbn |Φi
(4.59)
Handy Trick: There is no reason to carryout contractions that we know in advance will be zero. For example: all contractions between two creation operators r † s† , or two annihilation operators rs, any contraction between a pair ai† (instead of i† a), and any contraction between a pair i† a (instead of ai† ). Wick’s Theorem (simplified version)
is the sum of all nonzero contractions. This simplified version of Wick’s theorem is already very powerful. We shall now do a few applications.
4.6
Hartree-Fock Energy Expression
The Hartree-Fock energy expression is just the expectation value, ˆ . EHF = hΦHΦi Let us evaluate this using Wick’s theorem. The 1-electron part is, X (1) † ˆ EHF = hr|h|sihΦ|r s|Φi .
(4.60)
(4.61)
r,s
Since there is only one nonzero contraction, +--+ + |+ | < r s > = < r s > then it follows that, (1)
EHF =
X r,s
=
X
ˆ hr|h|siδ r,s nr hii .
(4.62)
i
and we are following the “Fortran index convention” so that i is a sum over occupied orbitals only. We may represent this term diagrammatically as,
Notice how the line corresponds to the contraction. The 2-electron part is, 1X (2) (rs||tu)hΦ|r †t† us|Φi . EHF = 2 There are two nonzero contractions,
(4.63)
56
CHAPTER 4. TIME-INDEPENDENT THEORY
+-------+ | +--+ | + + |+ |+ | |
= < r t u s >
0 intersections => plus
+-----+ | +--|-+ |+ |+ | | +
1 intersections => minus
It follows that, (2)
1X (rs||tu) (δr,s δr,u − δr,u δt,s ) ns nu 2 X = [(ii||jj) − (ij||ji)] .
EHF =
(4.64)
In order to represent this graphically we need to introduce an additional diagrammatic convention, namely
The “dot form” is that of Hugenholtz while the “dashed-line form” (sometimes a wavy line is used (2) instead) is that of Feynman. Thus EHF is represented by,
The left-hand Feynman diagram is called a bubble diagram while the right-hand Feynman diagram is an oyster. The (−1)l+h rule works for the Feynman but not for the Hugenholtz form, so we will generally stick to the Feynman form. Also notice how the lines solid lines in the Feynman form correspond to the contractions obtained in applying Wick’s theorem. This is no accident, but the (1) (2) full explanation will have to wait until later. Putting it altogether EHF and EHF gives the familiar Hartree-Fock energy expression, EHF =
X i
hii +
X
[(ii||jj) − (ij||ji)] .
(4.65)
Notice that this derivation is much easier than the traditional derivation using the Slater-Condon rules, because second-quantization concisely summarizes these rules.
57
4.7. BRILLOUIN’S THEOREM
4.7
Brillouin’s Theorem
Theorem (Brillouin’s) Let Φ be the HF wave function and Φai = a† iΦ be a singly excited determinant (i.e., a “singles”), then ˆ a i = fi,a , hΦ|H|Φ (4.66) i where, fr,s = hr,s +
X
[(rs||jj) − (rj||js)] ,
(4.67)
is the Fock operator. Of course, the off-diagonal matrix elements of the Fock operator are zero for canonical HF orbitals. Exercise: Prove this theorem. The solution for the two electron part is given in Appendix 6.1. We can represent Brillouin’s theorem diagrammatically by,
The capping lines (dotted) are shown explicitly. They are needed for the (−1)h+l -rule in order to get the right signs. There is also a notation to indicate that everything takes place at zero time. This means that there are no cuts to be made giving energy denominators. In practice, it is easy to imagine making implicit both capping lines and the zero-time indication. Incendently, the second to last diagram is a bubble diagram while the last diagram is half an oyster.
4.8
CIS CIS is to excited states what HF is to ground states.
The minimal method for treating excited sates is the configuration interaction singles (CIS) method. The variational principle is applied to a trial function of the form ΨI = ΦC0 +
X
a† iΦCia .
(4.68)
i,a
Minimizing E=
ˆ Ii hΨI |H|Ψ hΨI |ΨI i
(4.69)
58
CHAPTER 4. TIME-INDEPENDENT THEORY
gives (a bit symbolically), "
ˆ ˆ bj i hΦ|H|Φi hΦ|H|Φ ˆ ˆ bi hΦai |H|Φi hΦai |H|Φ j
#
C0 Cjb
!
= EI
C0 Cia
!
.
(4.70)
Exercise: Use Wick’s theorem to show that the CIS matrix is, "
EHF fj,b fa,i Aia,jb + δi,j δab EHF
#
C0 Cjb
!
= EI
C0 Cia
!
.
(4.71)
ˆ ˆ b i was evaluated in Sec. 4.7. The The h|H|Φi was evaluated in Sec. 4.6. The hΦ|H|Φ j ˆ b i is given in Appendix 6.2. We can represent the solution for the solution for hΦai |H|Φ j one-electron part as,
Notice how we now have a disconnected diagram. However the (−1)h+l -rule still works and there is no time-ordering problem because everything is at the same time. The diagrams for the two-electron part are,
59
4.8. CIS
Putting it all together gives,
The elements fa,i = fj,b = 0 if we work with canonical HF orbitals (Brillouin’s theorem). Thus the excitation problem is completely decoupled from the ground state and we can write, ~ I = ωI C ~I , AC
(4.72)
ωI = EI − EHF .
(4.73)
where The A matrix can be directly determined using Wick’s rules, but it is also interesting to determine it using the relation, ˆ − EHF |Φb i Aia,jb = hΦai |H j h i † † ˆ = hΦ|i a H, b j |Φi .
(4.74)
The commutator also has the very interesting property that it reduces the number of creation and annihilation operators by two. Thus, h
ˆ b† j h,
i
= =
X
X
h
hp,q p† q, b† j
i
hp,q p† qb† j − b† jp† q
60
CHAPTER 4. TIME-INDEPENDENT THEORY = = =
and
X
hp,q δq,b p† j − p† b† qj − δj,pb† q + b† p† jq
X
hp,bp† j −
X
hp,q δq,b p† j − δj,pb† q X
hj,p b† q ,
(4.75)
ˆ b i = δi,j ha,b − δa,b hj,i . hΦai |h|Φ j
(4.76)
Aia,jb = δi,j δa,b (ǫa − ǫi ) + (ia||bj) − (ij||ba) .
(4.77)
In a similar manner, one finds,
Some additional insight into CIS may be gained by restricting ourselves to a two-orbital model. In a two-orbital model, we only consider transitions from orbital i to orbital a, that is i → a (i ↑→ a ↑) and ¯i → a ¯ (i ↓→ a ↓). The CIS matrix is then 2 × 2, "
ǫa − ǫi + (ia||ai) − (ii||aa) (ia||ai) (ia||ia) ǫa − ǫi + (ia||ai) − (ii||aa)
#
Cia C¯ia¯
!
=ω
Cia C¯ia¯
!
.
(4.78)
There are two solutions. The triplet solution is, ωT = ǫa − ǫi − (ii||aa) 1 C¯ia¯ = −Cia = √ . 2
(4.79)
The singlet solution is, ωS = ǫa − ǫi + 2(ia||ai) − (ii||aa) 1 C¯ia¯ = +Cia = √ . 2
4.9
(4.80)
MP2 Back to 1st order RSPT!
Møller-Plesset perturbation theory [16] is RSPT taking the Fock operator as zero-order Hamiltonian, X ˆ (0) = H fr,s r † s , (4.81) r,s
where,
SCF fr,s = hr,s + vr,s ,
and, SCF vr,s =
X i
is the self-consistent field. Diagrammatically,
[(rs||jj) − (rj||js)] ,
(4.82) (4.83)
61
4.9. MP2 The perturbation,
X 1 X SCF † (rq||sp)r †s† pq − vr,s r s, Vˆ = 2 p,q,r,s r,s
(4.84)
is known as the fluctuation potential. Diagrammatically speaking, the fluctuation potential removes all bubble and (whole and half) oyster terms but otherwise works just like a normal electron repulsion (dashed or wavy) line. Normally we use canonical HF orbitals so that the Fock operator is diagonal, X
ˆ (0) = H
ǫr r † r ,
(4.85)
r
The MP2 approach is second-order Møller-Plesset perturbation theory. In order to find the MP2 energy we must first find the first-order energy and wave function. The ground state of the zero-order Hamiltonian is just the HF determinant. So, ˆ E (0) + E (1) = hΦ|H|Φi = EHF .
(4.86)
From RSPT, the first-order part of the wave function is, (1)
Ψ0 =
X
(1) hI
ΨI
I6=0
(0)
|Vˆ |0(0) i
(0)
(0)
E0 − EI
.
(4.87)
The zero-order excited states can be classified by order of excitation, reference: singles: doubles: triples: where,
Φ ω=0 a a Φi ωi = ǫa − ǫi ab Φab ω = ǫa + ǫb − ǫi − ǫj , ij ij abc abc Φijk ωijk = ǫa + ǫb + ǫc − ǫi − ǫj − ǫk etc. (0)
(0)
ωI = EI − E0 ,
(4.88)
(4.89)
is the zero-order excitation energy. A key observation is that Vˆ is a two-electron operator. This means that the only excitations that can contribute to the first-order part of the wave function are at most doubly excited. However hΦ|Vˆ |Φai i consists diagrammatically of only bubbles and half-oysters, which have been cleverly removed by the fluctuation potential. Exercise: Evaluate hΦ|ˆ v |Φab ij i. The answer is given in Appendix 6.3. Diagrammatically the answer is,
62
CHAPTER 4. TIME-INDEPENDENT THEORY Exercise: Show that,
1 X a,b (ia||jb) − (ib||ja) , Φi,j 2 ǫa + ǫb − ǫi − ǫj
(4.90)
1 X (ia||jb)(jb||ia) − (ib||ja)(ja||ib) . 2 ǫa + ǫb − ǫi − ǫj
(4.91)
(1)
Ψ0 =
and that the second-order energy is, (2)
E0 =
A tricky point is the appearance of a factor of 1/2 which is needed to avoid overcounting b,a since Φa,b i,j = Φj,i . The wave function may be represented diagrammatically as,
Notice how exchange is handled. The corresponding energy diagrams are,
Energy diagram rules work pretty much the same way as in the case of the RSPT diagrams. The energy cuts give denominators which are differences of orbital energies with up-going (particle) orbital energies being subtracted from downgoing (hole) orbital energies. Complications arise because of factors of 2 and 1/2 depending upon type of diagram and how spin is or is not treated. If spin-orbitals are used (i.e., no explicit treatment of spin) there is a prefactor of 1/2 for Goldstone diagrams having a vertical mirror plane. For Hugenholtz diagrams, a factor of 2−k should be introduced for each pair of equivalent lines. Equivalent lines begin and end with the same interaction point. In the event of confusion, the best thing is to work out terms directly from RSPT and Wick’s theorem. It is always worth taking the extra time to make sure you really understand the conventions and that your formulae are correct!
Chapter 5 Time-Dependent Theory Real magic is when you understand each step but the final result remains just as amazing. At this point, you have learned the practical aspects of diagrammatic MBPT. The purpose of this final chapter is to complete your understanding of diagrammatic MBPT by showing how it arises from time-dependent perturbation theory and the time-dependent form of Wick’s theorem.
5.1
Wick’s Theorem (full version)
A simplified version of Wick’s theorem has already been given in Sec. 4.5. It was simplified version in two ways: (i) it was restricted to the time-independent problem and (ii) it was restricted to expectation values with respect to the physical vacuum. The full version of Wick’s theorem [17] is given here, without proof. (A proof may be found in Ref. [13] pp. 83-92.) Wick’s theorem concerns arbitrary time-ordered products of creation and annihilation operators. For example, T [a†int (t1 )i†int (t2 )bint (t3 )jint (t4 )] . (5.1)
The time-ordering operator was defined in Sec. 2.3. The definition has to be refined to specify that the creation and annihilation operators are to be anticommuted until the product is time-ordered. Thus if t2 > t1 > t3 > t4 , then, T [a†int (t1 )i†int (t2 )bint (t3 )jint (t4 )] = −i†int (t2 )a†int (t1 )bint (t3 )jint (t4 ) .
(5.2)
We will want to express time-ordered products of creation and annihilation operators in terms of normally ordered operators and contractions. Creation and annihilation operators become time-dependent in the interaction representation. How they do this is not quite obvious. A general operator in the interaction representation, ˆ ˆ −ifˆ Aˆint (t) = eif Ae ,
(5.3)
satisfies the equation of motion, i
h i h i ∂ ˆ ˆ fˆ (t) . Aint (t) = Aˆint (t), fˆ = A, int ∂t
(5.4)
For convenience, the zero-order Hamiltonian has been chosen to be the Fock operator, fˆ =
X
ǫr r † r ,
r
63
(5.5)
64
CHAPTER 5. TIME-DEPENDENT THEORY
ˆ (0) . Now suppose Aˆ is a creation or an but Eq. (5.4) holds generally when fˆ is replaced with H annihilation operator. It is easy to show that h
Hence i
i
r, fˆ = ǫr r ,
(5.6)
∂ rint (t) = ǫr rint (t) , ∂t
(5.7)
rint (t) = e−iǫr t r .
(5.8)
† rint (t) = e+iǫr t r † .
(5.9)
which has the solution, Naturally, It is interesting to see what this means for the second-quantized form of an operator in the interaction representation. Consider an arbitrary time-independent one-electron operator, Aˆ = Its interaction representation is, Aˆint (t) = =
X
X
X
Ar,s ir † s .
(5.10)
† Ar,s rint (t)sint (t)
Ar,s e+iǫr t r † se−iǫs t .
(5.11)
The representation of an arbitrary two-electron operator is similar. An operator is normally ordered if all particle annihilation operators are placed to the right of all particle creation operators and all hole creation operators are placed to the right of all particle annihilation operators. The normal ordering operator anticommutes creation and annihilation operators by anticommuting them until they are in normal order. For example, N [a†int (t1 )i†int (t2 )bint (t3 )jint (t4 )] = +bint (t3 )jint (t4 )a†int (t1 )i†int (t2 ) = −bint (t3 )jint (t4 )i†int (t2 )a†int (t1 ) . (5.12) Normal ordering is interesting because expectation values of normally-ordered operators with respect to the physical vacuum vanish. Let Uˆ and Vˆ be two creation and/or annihilation operators. Then their contraction is just the difference between the time-ordered and the normally-ordered products, ˆ Vˆ ] , Uˆ (1) Vˆ (1) = T [Uˆ Vˆ ] − N [U
(5.13)
Exercise: Show that (1)
(1)
rint (t1 )sint (t2 ) = 0 †(1)
†(1)
(1)
†(1)
†(1)
(1)
rint (t1 )sint (t2 ) = 0 rint (t1 )sint (t2 ) = e+iǫs (t2 −t1 ) θ(t1 − t2 )δr,s n ¯s
rint (t1 )sint (t2 ) = e+iǫs (t1 −t2 ) θ(t1 − t2 )δr,s ns ,
(5.14)
n ¯ s = 1 − ns .
(5.15)
where,
65
5.2. DIAGRAMS The nonzero contractions are elements of the (zero-order) Green’s function (or operator), † iG(0) r,s (t1 , t2 ) = hΦ|T [rint (t1 )sint (t2 )]|Φi
= e+iǫs (t2 −t1 ) δr,s [¯ ns θ(t1 − t2 ) − ns θ(t2 − t1 )] .
Exercise: Show that G(0) r,s (ω) =
(5.16)
δr,s , ω − ǫs + i(ns − n ¯ s )η
(5.17)
where η = 0+ is a convergence factor needed to make the Fourier transform well defined. We may now finally state, Theorem (Wick) The time-ordered product of second-quantized operators is the sum over all possible contractions of the normal-ordered product. For example, (0)
T [a†int (t1 )i†int (t2 )bint (t3 )jint (t4 )] = bint (t3 )jint (t4 )a†int (t1 )i†int (t2 ) − iGb,a (t3 , t1 )jint (t4 )i†int (t2 ) (0)
(0)
(0)
− iGj,i (t1 − t2 )bint (t3 )a†int (t1 ) − iGb,a (t3 , t1 )iGj,i (t1 − (5.18) t2 ) .
Corollary The expectation value of the time-ordered product of second-quantized operators with respect to the physical vacuum is just the sum over all possible nonzero contractions. It is the corollary which is so very useful in diagrammatic MBPT.
5.2
Diagrams Coming full cycle
We may now put together all of the elements up to this point to see the true meaning of the diagrams used in MBPT. The presentation here is just a sketch but should be sufficient to comprehend the fundamental concepts. In Sec. 2.4 we have already seen the linked-cluster theorem without proof in the context of ordinary RSPT. The original proof was actually given by Goldstone in the context of MBPT [9]. If you think about it, this almost a bit surprising since each RSPT node will break up into several separate pieces, leading to additional unlinked diagrams. However Goldstone also showed that these additional diagrams are also cancelled by the denominator in the Gell-Mann and Low theorem. The energy expression is then, (0)
(0)
E0 = E0 + hΨ0 |Vˆint (0)
∞ X
n=0
(0) Uˆ (n) (0, −∞)|Ψ0 iL ,
(5.19)
where the subscript L means that only linked diagrams are to be taken into account. Now, for the sake of understanding, let us restrict ourselves to a one-particle system and look only at the n = 2 term, (3)
E0
=
(−i)2 2!
Z
0
−∞
dt1
Z
0
−∞
(0)
(0)
dt2 hΨ0 |Vˆint (0)T [Vˆint (t1 )Vˆint (t2 )]|Ψ0 iL
66
CHAPTER 5. TIME-DEPENDENT THEORY Z
Z
0 (−i)2 0 (0) (0) dt1 dt2 hΨ0 |T [Vˆint (0)Vˆint (t1 )Vˆint (t2 )]|Ψ0 iL = 2! −∞ −∞ Z 0 Z 0 (−i)2 X (0) (0) † = dt2 hΨ0 |T [rint dt1 (0)sint (0)p†int (t1 )qint (t1 )t†int (t2 )uint (t2 )]|Ψ0 iL Vr,s Vp,q Vt,u 2! −∞ −∞
= − .
X
Vr,s Vp,q Vt,u
Z
0
dt1
−∞
Z
t1
(0)
(0)
† dt2 hΨ0 |rint (0)sint (0)p†int (t1 )qint (t1 )t†int (t2 )uint(t2 )|Ψ0 iL
−∞
(5.20)
By Wick’s theorem and the linked cluster theorem we have, (3) E0
= − −
X
X
Vr,s Vp,q Vt,u
Vr,s Vp,q Vt,u
Z
Z
0
−∞
0
−∞
Z
dt1
dt1
Z
t1 −∞
t1
(0)
(0) dt2 iG(0) s,q (t1 , 0)iGu,p (t2 , t1 )iGr,t (0, t2 ) (0)
−∞
(0) dt2 iG(0) r,p (0, t1 )iGq,t (t1 , t2 )iGu,s (t2 , 0) .
(5.21)
We can now draw diagrams using the convention that each iG(0) (t1 , t2 ) becomes an arrow directed from t2 to t1 . (Usually we draw the diagrams to define what we meant by linked cluster, but here we are trying to show how the diagrams follow from the equations. They are in fact rather inseperable.) This gives the familiar third order energy diagrams that we have already seen,
The first term in Eq. (5.21) corresponds to diagram (a) while the second term corresponds to diagram (b). There is a negative sign which comes from the factor (−1)l . We will now put in explicit expressions for the Green’s functions, thus providing the other (−1)h factor, (3) E0
= − −
X
= − +
Z
Z
Z
0
−∞ Z 0
Va,j Vi,j Va,i
Va,i Va,b Vb,i
dt1
dt1
−∞
0
−∞
Z
0
−∞
Va,j Vi,j Va,i
Va,i Va,b Vb,i
X
X
Vr,s Vp,q Vt,u
Vr,s Vp,q Vt,u
X
X
= − +
X
Z
−∞
0
−∞
Z
t1
−∞
t1
−∞
t1
−∞
dt2 (−)δs,q nq e+iǫq t1 (−)δu,p np e+iǫp (t2 −t1 ) (+)δr,t n ¯ t e−iǫt t2
dt2 δr,p n ¯ p e−iǫp t1 δq,t n ¯ t e+iǫt (t1 −t2 ) (−)δu,s e+iǫs t2
−∞ Z t1
dt1
dt1 0
Z
Z
dt2 e+iǫj t1 e+iǫi (t2 −t1 ) e−iǫa t2
dt2 e−iǫa t1 e+iǫb (t1 −t2 ) e+iǫi t2
dt1 ei(ǫj −ǫi)t1 i(ǫb −ǫa )t1
dt1 e
Z
Z
t1
−∞
t1 −∞
dt2 ei(ǫi −ǫa )t2
dt2 ei(ǫi +ǫb )t2 .
(5.22)
Since we are doing adiabatic perturbation theory (and to make the integrals converge) we introduce convergence factors with η = 0+ , (3)
E0
= −
X
Va,j Vi,j Va,i
Z
0
−∞
dt1 ei(ǫj −ǫi+iη)t1
Z
t1
−∞
dt2 ei(ǫi −ǫa +iη)t2
67
5.3. SUMMARY OF CONVENTIONS +
X
= + =
X
Z
0
−∞
Z
dt1 ei(ǫb −ǫa +iη)t1
Z
t1
−∞
dt2 ei(ǫi +ǫb +iη)t2
1 ei(ǫi −ǫa +iη)t1 i(ǫi − ǫa + iη) −∞ Z 0 X 1 ei(ǫi +ǫb +iη)t1 dt1 ei(ǫb −ǫa +iη)t1 Va,i Va,b Vb,i i(ǫi + ǫb + iη) −∞ X Va,j Vi,j Va,i Z 0 dt1 ei(ǫj −ǫa +i2η)t1 − i(ǫi − ǫa + iη) −∞ X Va,i Va,b Vb,i Z 0 dt1 ei(ǫi −ǫa +i2η)t1 i(ǫi + ǫb + iη) −∞ X X Va,i Va,b Vb,i Va,j Vi,j Va,i − . (ǫi − ǫa + iη)(ǫj − ǫa + i2η) (ǫi + ǫb + iη)ǫi − ǫa + i2η)
= − +
Va,i Va,b Vb,i
Va,j Vi,j Va,i
0
dt1 ei(ǫj −ǫi+iη)t1
(5.23)
This gives us the usual expression for the third-order energy diagrams when we take the limit ǫ → 0.
5.3
Summary of Conventions
Several different styles of diagrams exist in the literature. It is convenient to give them a name according to how compressed they are: 1/r12 \ time Dashed lines Points
Not ordered Feynman Abrikosov
Ordered Goldstone Hugenholtz
Abrikosov diagrams are the most compressed while Goldstone diagrams are the most explicit. Brandow diagrams represent each Hugenholtz diagrams by one of the corresponding Goldstone diagrams.
68
CHAPTER 5. TIME-DEPENDENT THEORY
Chapter 6 Appendices 6.1
Evaluation of hΦ|ˆ v|Φaii Using Wick’s Theorem
Evaluate hΦ|ˆ v |Φai i where
Φai = a† iΦ
(6.1)
is a single excitation. (This integral is basic to Brillouin’s theorem.) As hΦ|ˆ v |Φai i =
1X (rs||tu)hΦ|r †t† usa† i|Φi , 2
(6.2)
we see that we must evaluate hΦ|r † t† usa† i|Φi. By using the method of connecting lines we have the following nonzero contractions, + + + < r t u s a i > +-----+ +-+ | +--|-|-|--+ |+ |+ | | |+ | = < r t u s a i > +---+ +-----|-+ | | +--|-|-|--+ |+ |+ | | |+ | + < r t u s a i >
+--+ +-+ +--|--|-|-|--+ |+ |+ | | |+ | + < r t u s a i >
3 intersections => minus (1)
4 intersections => plus (2)
4 intersections => plus (3)
69
70
CHAPTER 6. APPENDICES +----+ | +-|-+ +--|--|-|-|--+ |+ |+ | | |+ | + < r t u s a i >
5 intersections => minus (4)
Hence + + + < r t u s a i >
+--+ +--+ +-+ |+ | |+ | | |+ = - r u t i s a
(1)
+--+ +--+ |+ | |+ | + r s t i
(2)
+-+ | |+ u a
+--+ +--+ +-+ |+ | |+ | | |+ + r i t u s a
(3)
+--+ +--+ |+ | |+ | - r i t s
(4)
+-+ | |+ u a
We have that, hΦ|r † t† usa† i|Φi = −δr,u nu δt,i δs,a + δr,s ns δt,i δu,a + δr,i δt,u nu δs,a − δr,i δt,s ns δu,a .
(6.3)
So 1X (rs||tu) [−δr,u nu δt,i δs,a + δr,s ns δt,i δu,a + δr,i δt,u nu δs,a − δr,i δt,s ns δu,a ] 2 i 1h X − (ja||ij) + (jj||ia) + (ia||jj) − (ij||ja) = 2 X X = (ia||jj) − (ij||ja) = Ji,a − Ki,a .
hΦ|ˆ v|Φai i =
(6.4)
ˆ B 6.2. EVALUATION OF hΦA I |V |ΦJ i USING WICK’S THEOREM
6.2
71
v |Φbj i Using Wick’s Theorem Evaluation of hΦai|ˆ
Evaluation of hΦai |ˆ v|Φbj i. + + + + +--------------+ | +---------+ | | | +-----+ | | | | | +-+ | | | |+ | |+|+| | |+| =
0 intersections => plus
(1)
+--------------+ | +---------+ | | | +---+ | | | | +-|-+ | | | |+ | |+|+| | |+| +
1 intersections => minus
(2)
+--------------+ | +-+ +-+ +-+ | |+ | |+|+| | |+| +
0 intersections => plus
(3)
+--------------+ | +---+ | | +-+ +-|-+ | | |+ | |+|+| | |+| +
1 intersections => minus
(4)
+--------------+ | +-----+ | | +-|-+ +-|-+ | |+ | |+|+| | |+| +
2 intersections => plus
(5)
+--------------+ | +---+ | | +-|-+ | +-+ | |+ | |+|+| | |+| +
1 intersections => minus
(6)
72
CHAPTER 6. APPENDICES
+---------+ +--|-------+ | | | +-----|-|-+ | | | +-+ | | | |+ | |+|+| | |+| +
3 intersections => minus
(7)
+---------+ +----|---+ | | +-|---|---+ | | | | +-|-+ | | |+ | |+|+| | |+| +
4 intersections => plus
(8)
+--------+ | +-|-----+ | +---|-|---+ | | | +-|-|-+ | | |+ | |+|+| | |+| +
5 intersections => minus
(9)
+----------+ | +---|---+ | +---|---|-+ | | | +-|-+ | | | |+ | |+|+| | |+| +
4 intersections => plus
(10)
+-------+ +------|-+ | | +-+ | | +-+ | |+ | |+|+| | |+| +
1 intersections => minus
(11)
+--------+ | +---|-----+ | +-|-+ | +-+ | |+ | |+|+| | |+| +
2 intersections => plus
(12)
+----------+ | +---|---+ | +-+ | +-|-+ | |+ | |+|+| | |+| +
2 intersections => plus
(13)
ˆ B 6.2. EVALUATION OF hΦA I |V |ΦJ i USING WICK’S THEOREM
+----------+ | +-----|---+ | +-|-+ +-|-+ | |+ | |+|+| | |+| +
3 intersections => minus
Hence + + + +
=
+--+ +-+ +--+ |+ | | |+ |+ | i j a b r s
+--+ |+ | t u
(1)
-
+--+ +-+ +--+ |+ | | |+ |+ | i j a b r u
+--+ |+ | t s
(2)
+
+--+ +-+ +--+ |+ | | |+ |+ | i j a r t u
+--+ |+ | s b
(3)
-
+--+ +-+ +--+ |+ | | |+ |+ | i j a r t s
+--+ |+ | u b
(4)
+
+--+ +-+ +--+ |+ | | |+ |+ | i j a t r s
+--+ | |+ u b
(5)
-
+--+ +-+ +--+ |+ | | |+ |+ | i j a t r u
+--+ | |+ s b
(6)
-
+--+ +-+ +--+ |+ | | |+ |+ | i s a b r j
+--+ |+ | t u
(7)
+
+--+ +-+ +--+ |+ | | |+ |+ | i u a b r j
+--+ |+ | t s
(8)
73
(14)
74
CHAPTER 6. APPENDICES
-
+--+ +-+ +--+ |+ | | |+ |+ | i u a b r s
+--+ |+ | t j
(9)
+
+--+ +-+ +--+ |+ | | |+ |+ | i s a b r u
+--+ |+ | t j
(10)
-
+--+ +-+ +--+ |+ | | |+ |+ | i u a r t j
+--+ | |+ s b
(11)
+
+--+ +-+ +--+ |+ | | |+ |+ | i s a r t j
+--+ | |+ u b
(12)
+
+--+ +-+ +--+ |+ | | |+ |+ | i s a r t j
+--+ | |+ u b
(13)
-
+--+ +-+ +--+ |+ | | |+ |+ | i s a t r j
+--+ | |+ u b
(14)
So
= − + − + − − + − +
hΦai |ˆ v|Φbj i 1X (rs||tu)δi,j δa,b δr,s ns δt,u nu (1) 2 1X (rs||tu)δi,j δa,b δr,u nu δt,s ns (2) 2 1X (rs||tu)δi,j δa,r δt,u nu δs,b (3) 2 1X (rs||tu)δi,j δa,t δt,s ns δu,b (4) 2 1X (rs||tu)δi,sδa,t δr,s ns δu,b (5) 2 1X (rs||tu)δi,j δa,t δr,u nu δs,b (6) 2 1X (rs||tu)δi,sδa,b δr,j δt,u nu (7) 2 1X (rs||tu)δi,uδa,b δr,j δt,s ns (8) 2 1X (rs||tu)δi,uδa,b δr,s ns δt,j (9) 2 1X (rs||tu)δi,sδa,b δr,u nu δt,j (10) 2
ˆ B 6.2. EVALUATION OF hΦA I |V |ΦJ i USING WICK’S THEOREM 1X (rs||tu)δi,uδa,r δs,bδt,j (11) 2 1X (rs||tu)δa,tδs,b δr,j δi,u (12) + 2 1X + (rs||tu)δi,sδa,r δu,b δt,j (13) 2 1X (rs||tu)δi,sδa,t δu,b δr,j (14) . − 2
75
−
or
= − + − + − − + − + − + + −
hΦai |ˆ v |Φbj i 1X δi,j δa,b (kk||ll) (1) 2 1X (kl||lk) (2) δi,j δa,b 2 1X (ab||kk) (3) δi,j 2 1X δi,j (ak||kb) (4) 2 1X (kk||ab) (5) δi,j 2 1X δi,j (kb||ak) (6) 2 1X (ji||kk) (7) δa,b 2 1X (jk||ki) (8) δa,b 2 1X δa,b (kk||ji) (9) 2 1X (ki||jk) (10) δa,b 2 1 (ab||ji) (11) 2 1 (jb||ai) (12) 2 1 (ia||jb) (13) 2 1 (ji||ab) (14) . 2
(6.5)
(6.6)
Donc hΦai |ˆ v |Φbj i =
i X 1 hX (kk||ll) − (kl||lk) 2 + δi,j [Ja,b − Ka,b ) − δa,b (Jj,i − Kj,i) − (ab||ji) + (jb||ai) .
= δi,j δa,b
(6.7)
76
6.3
CHAPTER 6. APPENDICES
Evaluation of hΦ|ˆ v|Φab ij i Using Wick’s Theorem
To evaluate, hΦ|ˆ v |Φab ij i = we must evaluate + + + + +-----------------+ | +--------+ | | +--|------+ | | | | | +-+ | | | |+ |+ | | |+ | |+ | =
(1)
+-----------------+ | +------+ | | +----|----+ | | | | +-|-+ | | | |+ |+ | | |+ | |+ | +
(2)
+--------------+ | +------+ | +--|----|----+ | | | | +-|-+ | | | |+ |+ | | |+ | |+ | +
(3)
+------------+ | +---------|----+ | | +------|-+ | | | | +-+ | | | |+ |+ | | |+ | |+ | +
(4)
So
1X (pq||rs)hp†r † sqb† ja† ii , 2
(6.8)
6.3. EVALUATION OF hΦ|Vˆ |ΦAB IJ i USING WICK’S THEOREM
77
+ + + +
+-+ +-+ +-+ +-+ |+| |+| | |+ | |+ = - p i r j q b s a
(1)
+
+-+ +-+ +-+ +-+ |+| |+| | |+ | |+ p i r j q a s b
(2)
-
+-+ +-+ +-+ +-+ |+| |+| | |+ | |+ p j r i q a s b
(3)
+
+-+ +-+ +-+ +-+ |+| |+| | |+ | |+ p j r i q b s a
(4)
Hence
= = + − + = =
hΦ|ˆ v |Φab ij i 1X (pq||rs)hp†r † sqb† ja† ii 2 1X (pq||rs)δp,iδr,j δq,b δs,a (1) − 2 1X (pq||rs)δp,iδr,j δq,a δs,b (2) 2 1X (pq||rs)δp,j δr,i δq,a δs,b (3) 2 1X (pq||rs)δp,j δr,i δq,b δs,a 4) 2 1 1 1 1 − (ib||ja) + (ia||jb) − (ja||ib) + (jb||ia) 2 2 2 2 (ia||jb) − (ib||ja) .
(6.9)
78
CHAPTER 6. APPENDICES
6.4
Rules for Goldstone Energy Diagrams
Summary of rules for Møller-Plesset perturbation theory energy diagrams using the Goldstone conventions (Ref. [15] pp. 362-368). • Rule 0 Draw all topologically distinct linked diagrams.
• Rule 1
• Rule 2 Each horizontal cut,
contributes a denominator of, X
hole
ǫi −
X
particle
ǫa = (ǫi + ǫj + · · ·) − (ǫa + ǫb + · · ·) .
(6.10)
• Rule 3 There is a prefactor of (−1)h+l where h is the number of hole lines and l is the number of loops. • Rule 4 Sum over all particle and hole indices. • Rule 5 Diagrams with a vertical mirror plane perpendicular to the plane of he paper are to be multiplied by a factor of 1/2.
6.5
Rules for Hugenholtz Energy Diagrams
Summary of rules for Møller-Plesset perturbation theory energy diagrams using the Hugenholtz conventions (Ref. [15] pp. 362-368). • Rule 0 Draw all topologically distinct linked diagrams.
6.5. RULES FOR HUGENHOLTZ ENERGY DIAGRAMS
79
• Rule 1
• Rule 2 Each horizontal cut,
contributes a denominator of, X
hole
ǫi −
X
particle
ǫa = (ǫi + ǫj + · · ·) − (ǫa + ǫb + · · ·) .
(6.11)
• Rule 3 There is a prefactor of (−1)h+l where h is the number of hole lines and l is the number of loops. Calculate it by expanding the diagram into a representative Goldstone diagram. (It does not matter which one.) • Rule 4 Sum over all particle and hole indices. • Rule 5 There is a prefactor of 2−k where k is the number of equivalent lines—that is k is the number of pairs of lines that go in the same direction and start and end at the same dot.
80
CHAPTER 6. APPENDICES
Bibliography [1] M. E. Casida and D. P. Chong, Contribution of correlation and relaxation to generalized overlaps for outer-valence ionization, Chem. Phys. 133, 47 (1989), Erratum, ibid, 136, 489 (1989). [2] M. E. Casida and D. P. Chong, Physical interpretation and assessment of of the coulomb-hole and screened-exchange approximation for molecules, Phys. Rev. A 40, 4837 (1989), Erratum, ibid, 44, 6151 (1991). [3] M. E. Casida and D. P. Chong, Quasi-particle equation from the configuration-interaction (CI) wave-function method, Int. J. Quant. Chem. 40, 225 (1991). [4] M. E. Casida and D. P. Chong, Simplified green-function approximations: Further assessment of a polarization model for second-order calculation of outer-valence ionization potentials in molecules, Chem. Phys. 159, 347 (1991). [5] M. E. Casida, Generalization of the optimized effective potential model to include electron correlation: A variational derivation of the Sham–Schluter equation for the exact exchange-correlation potential, Phys. Rev. A 51, 2005 (1995). [6] C.-H. Hu, D. P. Chong, and M. E. Casida, The parameterized second-order Green function times screened interaction (pGW2) approximation for calculation of outer valence ionization potentials, J. Electron Spectr. 85, 39 (1997). [7] M. E. Casida, Correlated optimized effective potential treatment of the derivative discontinuity and of the highest occupied Kohn-Sham eigenvalue: A Janak-type theorem for the optimized effective potential method, Phys. Rev. B 59, 4694 (1999). [8] M. E. Casida, Propagator corrections to adiabatic time-dependent density-functional theory linear response theory, J. Chem. Phys. 122, 054111 (2005). [9] J. Goldstone, Derivation of the Bueckner many-body theory, Proc. Roy. Soc. (London) A239, 267 (1957). [10] J. O. Hirschfelder, W. B. Brown, and S. T. Epstein, Recent developments in perturbation theory, Adv. Quant. Chem. 1, 255 (1964). [11] F. L. Pilar, Elementary Quantum Chemistry, McGraw-Hill, New York, 1968. [12] M. Gell-Mann and F. Low, Bound states in quantum field theory, Phys. Rev. 84, 350 (1951). [13] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. 81
82
BIBLIOGRAPHY
[14] L. I. Schiff, Quantum Mechanics, 3rd Edition, McGraw-Hill, New York, 1949. [15] A. Szabo and N. S. Ostland, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, MacMillan Publishing Co., New York, 1982. [16] C. Møller and M. S. Plesset, Note on ann approximation treatment for many-electron systems, Phys. Rev. 46, 618 (1934). [17] G. C. Wick, The evolution of the collision matrix, Phys. Rev. 80, 268 (1950).
