Epidemiology

• Higher incidence (9 – 20/100,000 person years) and prevalence (156 – 291/100,000 people) in populations of North American and Northern European descent (Lancet 2012;380:1606)
• Incidence increased in industrialized countries and urban versus rural locations, suggestive of environmental triggers, such as improved sanitation, reduced exposure to childhood enteric infections and mucosal immune system maturation (Lancet 2012;380:1606)
• Bimodal age distribution with peaks at 15 – 30 years and 50 – 70 years (Lancet 2012;380:1606)
• Family history of inflammatory bowel disease, particularly that of a first degree relative (5.7 – 15.5%) and Ashkenazi Jewish descent (3 – 5x) show higher risk of disease development (Lancet 2012;380:1606)
• Gastrointestinal infections with Salmonella spp, Shigella spp and Campylobacter spp have twice the risk of developing ulcerative colitis postinfection (Lancet 2012;380:1606)
• M = F
• Former cigarette smoking is strong risk factor (Lancet 2017;389:1756)

• Almost always involves the rectum
  • Continuous pattern of involvement proximally to include up to the entire colon (pancolitis)
  • Rectal sparing can be seen, particularly after treatment (Lancet 2012;380:1606, Histopathology 2014;64:317, Am J Clin Pathol 2004;122:94)
• Patch of inflammation in the cecum, often involving the periappendiceal mucosa (cecal patch), can be present (Lancet 2012;380:1606, Histopathology 2014;64:317)
• Approximately 20% of patients will have inflammation in the terminal ileum (backwash ileitis)
  • Typically present in patients with pancolitis (Am J Surg Pathol 2005;29:1472)
• Focally enhanced gastritis can be seen in approximately 20% of pediatric patients (Pathology 2017;49:808)
• Extraintestinal manifestations:
  • Peripheral arthritis, seronegative
  • Ankylosing spondylitis or sacroiliitis
  • Erythema nodosum
  • Pyoderma granulosum
  • Primary sclerosing cholangitis (PSC)

Pathophysiology

• Not fully known but appears to be a complex multifactorial process involving an overwhelming T helper type 2-like immune response, leading to mucosal injury in response to gut microbial dysbiosis in genetically predisposed patients
• Proposed mechanisms include:
  • Damage to the colonic epithelial barrier due to dysregulation of epithelial tight junctions, which provide a physical barrier between the immune cells and the luminal microbes, leads to increased permeability (Lancet 2012;380:1606)
  • Colonic epithelium upregulation of antimicrobial peptides, known as beta defensins (Lancet 2012;380:1606)
  • Disruption in the homeostatic balance of the mucosal immunity and the enteric nonpathogenic bacteria, resulting in the patient’s aberrant immune response to the enteric commensal bacteria (Lancet 2012;380:1606, Front Microbiol 2018;9:2247)
  • Increased number of colonic epithelium activated and mature dendritic cells with increased stimulatory capacity (Lancet 2012;380:1606)
  • Increased expression of TLR4 by lamina propria cells and TLR4 polymorphism, which can alter susceptibility to enteric infections and tolerance to commensal bacteria (Lancet 2012;380:1606)
  • Disruption in the homeostatic balance between regulatory and effector T cells, leading to a nonclassic natural killer T cell production of IL5 and IL13, which have cytotoxic effects on epithelial cells, mediating an atypical Th2 response
    • IL13 can induce a positive feedback system on the natural killer T cells, leading to increased tissue injury (Lancet 2012;380:1606)
  • Increase in proinflammatory cytokines, chemoattractants such as CXCL8 and adhesion molecules such as MadCAM1 recruit increased leukocytes to the colonic mucosa (Lancet 2012;380:1606)
  • Other genetic risk loci include IL23 and IL10, JAK2 kinase pathway genes, hepatocyte nuclear factor 4α, CDH1 and laminin β1 (Lancet 2012;380:1606)

Etiology :

I propose a mechanism that involves the limbic system and that continuously maintains a state of stress equivalent to a situation of major stress (trauma) and therefore causes all the signs and symptoms of the disease. How does that happen?
Imagine the situation of this gazelle:

Its limbic system will use all means to save its life, by starting it will increase blood circulation in the muscles by dilating the arteries of the muscles (vasodilation), it will increase blood pressure to increase the pressure of perfusion, it will increase heart rate to increase blood flow per minute. This system will trigger other reactions at the level of other organs for example, in the liver and fat tissue it will cause an increase in the release of carbohydrates and fatty acids to bring the necessary energy to the muscles. It will give to this gazelle a deeper field of vision as well as many others possibilities with other systems (see physiopathology of stress) but this system will go further it will reduce the circulation in the internal organs to offer the muscles the blood flow necessary for survival (its survival is at stake the intestines and other organs can wait).

It will also stop the movement of the intestine  and the evacuation of the urinary system, because this is not the time to go to the bathroom!So no defecation and no urination. At the same time  for the same reason the appetite is inhibited  But while this gazelle is running it will see a lot of things it will see flowers, plants, streams, trees, the sun, as well as other animals…… in short it will see its usual environment in complete.Probably consciously it will record only the traumatic part of what happens to it, because it is concerned about its survival, but subconsciously all these elements will be recorded. As a result, it will consciously record a small portion of the information and a large portion of the information will be recorded subconsciously. If this gazelle succeeds in saving its life, it cannot help but record those moments of stress in his memories.

After this event this gazelle is condemned to live with the stress that is recorded and this state of stress will have a continuous effect because the environment of this gazelle’s life is filled with elements which unconsciously remind it of the stressful event and which trigger a major stress with all the resulting consequences. Inhibition of bowel motility will cause constipation. Abdominal and digestive artery vasoconstriction will result in functional ischemia whose intensity will determine the symptomatology. If the vasoconstriction is limited it will only cause slight clinical manifestations without a lesion visible macroscopically or microscopically at the level of the intestine it is the case of the irritable bowel syndrome.

If the vasoconstriction is more important, there will be ischemia creating a minor cellular suffering and especially in the segments where the intestine is in the remote areas of the irrigation network and especially without anastomosis ie in the areas vulnerable. 

This explains the frequency with which the rectum gets affected…

The post New treatment of inflammatory bowel disease appeared first on BORZUYA UNIVERSITY.

In this section, we consider that every object can be considered as a dot without volume.

The position

The motionis the history of the position of an object, the succession of the positions of this object over the time. To define the position of the object, we need a reference system in which we can give the position: coordinates. Several systems of reference exist and there are several ways to calculate the coordinates of objects. All of them are corrects but some are more convenient than the others. During a trip, you won’t give your position with regard to the sun, the same is true in physics.

Several coordinates are generally necessary to determine the exact position of one object. If you say that you are 10km from Paris, you give an information on your position but we are lacking at least one coordinate to determine your position. Usually you need one coordinate by dimension of the system.

One dimension

We choose one origin to the coordinate system, the zero point. It is convenient to choose the initial position of the object A as the origin but it is not mandatory. Next we choose a direction that will be the positive positions. In the opposite direction we have the negative positions. Still for convenience, the positive positions are placed in the direction we guess the object will move towards. Imagine that the object moves towards another object B placed at a distance d. The position vector indicates the distance between an object and the origin, and points towards the object with an arrow. The symbol for vectors is topped by an arrow pointing to the right. The position vector for B is

Where  is the unit vector in the direction x (the single direction in this problem).

Two dimensions

The second coordinate is usually orthogonal, perpendicular to the first coordinate to avoid a maximum of angle problems and to benefit the simplicity of the calculation for right triangles. Each coordinate has a direction.

The position vector is now defined by two components from which we can calculate its length if desired.

The addition sign between the two unit vectors is seen as “followed by” and not like the addition of two usual numbers. Another way to write coordinates is to put them in brackets. If we do this, then we don’t write the unit vectors.

This system of reference, the Cartesian system, is not the single one that can be used at two dimensions to determine the position of an object. We can also position the object from its distance r to the origin point and an angle θ from one axis. This reference system is called the polar system.

It is possible to determine the relation between the coordinates X;Y in the Cartesian system and r;θ in the polar system using the relation defining the cosine, the sinus and the tangent:

The unity vector  can thus be calculated.

or

We can also define the unit vector :

Both polar unity vectors depend thus upon θ that should thus be chosen conscientiously.

Three dimensions

A third coordinate is added in the reference systems. In the Cartesian system, we add a coordinate that is orthogonal to the two previous ones.

In the polar system, we need a second angle to determine the position of an object. We take the first from the axis x in the xy plane and the second angle is taken from the z axis in the zr plane of the object.

Again, the Cartesian system can be associated to the polar system.

The unit vectors are defined as

The post Chapter 3a: the motion- the position appeared first on BORZUYA UNIVERSITY.

On the other hand, we can measure a distance in meters or in miles, the weight in grams or in pounds. All these units have a well-defined value and can serve as reference units. While some countries as the UK or the USA use some different units, international conventions defined an international system of units (SI base units). Their biggest advantage is the simple relation between the units of different quantities. In SI, one millilitre of water occupies one cubic centimetre, weighs one gram, and requires one calorie of energy to heat up by one degree centigrade, which is one percent of the difference between its freezing point and its boiling point. In the American system, you will need to make huge calculations to calculate how much energy it takes to boil a room-temperature gallon of water because you can’t directly relate any of those quantities.

The International Bureau of Weights and Measures (French: Bureau international des poids et mesures) is an intergovernmental organization, established to maintain the International System of Units (SI) under the terms of the Metre Convention (Convention du Mètre, May 20^th 1875). The organisation is usually referred to by its French initialism, BIPM. Its role is to

• establish fundamental standards and scales for the measure of main physical quantities and to conserve the international prototypes;
• compare international standards with national standards;
• ensure the coordination of the corresponding techniques of measurement;
• measure and coordinate the measures of the fundamental, physical constants involved in the above activities.

SI Units

The definitions of the reference units are mainly made to give them a well-known and fixed value.

These definitions fix the speed of light c at 299792458 m/s et the permeability of the void μ₀ at 4π 10^-7 H/m exactly. They also sometimes require some precisions. For instance, we point out that the cesium atom is at rest, that the carbon atoms are not connected, are at rest and in their fundamental state.

Deriving units

By commodity, some units are the combination of the SI units to express frequently used units.

Finally, there are some quantities without specific unit names

We can point out a few specific units:

To put an end to this section, we will list the prefix of multiples of the SI units.

Dimensional analysis

The existence of the IS system means that all the other physical quantities have units that are homogeneous functions of these base units. A function is homogeneous if, making a scale change on all of its variables: x₁ → λ₁x₁, x₂ → λ₂x₂, x₃ → λ₃x₃, … the function itself changes of scale: f(λ₁x₁, λ₂x₂, λ₃x₃,…) = λ₁^α1 λ₂^α2 λ₃^α3…f(x₁, x₂, x₃,…). The units of any quantity, let’s call it Q_r is thus always expressed by a relation like

So if we know the units of one physical quantity, and admitting that there is a relation between this quantity and other variables, and knowing the units of those variables, we can guess the relation between the quantities.

For instance, we observe the swinging of one object attached to one string: a pendulum. The pendulum oscillates because it falls and it is restrained by the string. We want to determine the relation between the times the pendulum takes to make one oscillation, i.e. the period, to the parameters we guess as important: the mass M of the object, the length L of the string, and the gravity acceleration g that affects each object on the planet. The units of the variables are kg for the mass, m for the length, and m/s² for the acceleration. The period is a time and thus its unit is s. The relation between the variables and the period should be something like this:

As there is no kg at the left of the equality, and it is present at the right side of the equation with the exponent a, then we conclude that a=0. Looking at the seconds, their exponent is 1 at the left and -2b at the right, b is thus b=-1/2. Finally, there is no m at the left while it is present at the right side of the equation, thus 0=b+c. As we determined the value of b, we have that c=1/2 and that the global relation is

From our dimensional analysis, we determined that the mass of the object has no influence on the period of the oscillation of the pendulum. Note that we did not write the equal sign: the dimensional analysis doesn’t give the true law; it gives clues on the variables but there can still be numerical factors that can be determined experimentally. On the other hand, the dimensional analysis allows to identify wrong laws not respecting the units of the quantities.

Let’s analyse a second example: the period T of revolution of planets around the Sun. First we identify the important parameters involved in the problem: the mass M of the Sun, the distance R between the planet and the Sun, and a constant G giving the gravitational force. The units of the parameters are respectively kg, m and kg^-1m³s^-2. The period is given in seconds s. The law should be of the form T ~ M^aG^bR^c. For the units, we have the relation

The next step is to identify the exponent of each unit at the left and the right side of the equation:

The law is thus written

This relation is the expression of the Kepler’s law that describes the trajectory of planets of the solar system.

Scales and orders of magnitude

An experimental approach is to estimate the order of magnitude of variables that appear in physical processes. Either we measure the characteristics of one well known property of the matter, or we determine a transition zone between two models of description.

For instance, we can regroup the matter as solids, liquids and gases. One major difference between these three phases is their density, i.e. the mass of the matter for a given volume, given in kg/m³. We can thus regroup liquids as matter with a density with the order of magnitude around 10³kg/m³ (at T=293K, water: 1003kg/m³, olive oil: 910kg/m³, sulfuric acid: 1834kg/m³,…)while gases have a density of order 1 (at T=273K, air: 1.2kg/m³, CO₂: 1.98kg/m³, methane: 0.72kg/m³,…). Between solids and liquids, there is a factor 10 in density (at T=293K: iron: 7893kg/m³, copper: 8954kg/m³, gold: 19320kg/m³,…). We will want to determine the temperature at which a solid becomes liquid, i.e. its melting temperature.

As the interactions between particles of a solid differ from the interactions between particles in a gas, laws are not the same at the microscopic scale than in the astronomic scale, not because the interactions mysteriously disappear, but because we can neglect some interactions. For instance, imagine an interaction between particles that depends directly on the distance between two particles and one interaction that depends on the third power of the distance. If the distance is small, both interactions will have an effect, but as soon as the distance gets large, we can neglect the first interaction.

The post Chapter 2: quantities and units appeared first on BORZUYA UNIVERSITY.

“One of the noblest desire of the man is to know the laws ruling the Universe, and those who contributed to enlighten some of the mysteries were always admired by their peers; they appear as privileged, wearing on them the divine light, a through centuries the generations gaze upon their indelible work and rank them first amongst the glories of the humanity.”

From Achille Cazin (Hachette, 1881).

Physics can be defined as a science that studies the general properties of matter, space, and time, and establishes laws that describe natural phenomena. We will point out the word law first, and ask ourselves if we can clearly, and without ambiguity, define all the objects we are discussing, i.e. space, time and matter. Their definitions are (Oxford dictionaries)

• Space: A continuous area or expanse which is free, available, or unoccupied.
• Matter: Physical substance in general, as distinct from mind and spirit; (in physics) that which occupies space and possesses rest mass, especially as distinct from energy.
• Time: The indefinite continued progress of existence and events in the past, present, and future regarded as a whole.

It is not what we can call clearly defined terms and it doesn’t indicate any of the relations that may exist between them. If we go back in time, Isaac Newton gave its definitions (1687):

• The absolute space, which is without any relation with anything from the surroundings, is always unchanging and immobile. The relative space is any measure or mobile dimension of this space, which is defined with regards to its position with regards to objects that we consider as the immobile space…
• The quantity of matter is the measure that we obtain from its density and its volume…
• The absolute time, true and mathematic, is without relation to anything from the surroundings, and by its nature flows uniformly. The relative time, is any measure, accurate or not, of the duration of an event, that we use in place of the true time. I.e. the hour, day, month,…

It is difficult to avoid cross references and circular definitions. We will abandon the idea to define everything and accept the fact that some notions can make sense without being explicit. Physics is seeking for the laws that rule the reciprocal actions between one object and its surrounding. Enouncing those laws is not an easy job, and we have seen many changes in the history, as the understanding of the scientists evolved. For instance, L. Wouters gives in a school book of 1916 the law for the dilatation of bodies due to heat as

“The first effect that heat produces on bodies is to increase their volume, to dilate them”

This law is based upon its hypothesis on the nature of heat:

“Based on the modern hypothesis on the nature of heat, it results from the vibratory movement of the smallest molecules of the ponderable matter, and is transmitted via a fluid called aether. Aether is a subtle, perfectly elastic, substance that fills the intermolecular spaces as well as the so-called interplanetary voids. Heat is, at the end, a particular state of movement.”

Today, it is obvious that this law is false, mostly because of the notion of aether it is based on. The law itself is true with some exceptions. So, shall we give no definition nor law? Obviously we will, but laws should results from the simple relations obtained from experimentations: if I modify one physical quantity of my experiment, then another physical quantity changes, at this effect is repeated consistently if I repeat the experiment.

To end this introduction, I will extend the citation of Achille Cazin (Hachette, 1881) I opened the introduction with.

“No matter the study on which we work, there are some general rules that one must follow to avoid falling in annoying confusions. …

In physics, we observe all the circumstances around a natural phenomenon; we measure all the available quantities; we seek relations between theses quantities, and these relations are called the law of the phenomenon. When the phenomenon is too complex, and it looks like it is impossible to state one unique law, we modify the phenomenon, we make an experiment. Some circumstances seeming ancillary, are made negligible, and we observe the dominating quantities. From this experiment we obtain an approximate law, and by extension we seek the influence of the neglected circumstances, in which way they alter the law. Doing that, we find the limit law towards which tends the observed law when the circumstances become more and more negligible. Such a law is then used as a fundamental principle, seen as the temporary expression of a physical truth, temporary because one more accurate observation or a new phenomenon can modify the conclusions admitted as truth until now.”

The post Chapter 1: Elementary physics – Introduction appeared first on BORZUYA UNIVERSITY.

Keilin (1925) was studying the mosquitoes responsible of the malaria. With a spectromicroscope he looked at the legs of one mosquito and saw that they change of colour. At rest the muscles are oxidised but when they are excited the muscles consume the oxygen and are thus less oxidised. The colour is given by the cytochrome, a molecule that changes of colour in function of the oxidation.

This molecule belongs to the mitochondrion. The mitochondrion is a kind of bacteria hosted by eukaryotes.

They have their own membrane, composed of an inner membrane and of a porous outer membrane. The matrix is inside the inner membrane and the intermembrane space is also called cristae where the membrane is protruding. Similarly to the haemoglobin, the cytochrome is a protein that possesses a haem the iron of which can change of oxidation state. This property is used in the respiratory chain. Cytochrome belongs to the inner membrane of the mitochondrion along which it can move and it separate the glycolysis from the citric acid cycle. The cycle of the citric acid takes place in the matrix of the mitochondrion while glycolysis takes place outside of the mitochondrion.

Respiratory chain

The respiratory chain can be summarised by the following figure and table:

 In the inner membrane we find 4 complexes – NADH-ubiquinone reductase, succinate-ubiquinone reductase, ubiquinol-cytochrome c reductase and cytochrome oxidase – that make redox reactions and transfer electrons from one side of the membrane to the other side (sometimes protons instead of electrons). One reaction at one side of the membrane (matrix or intermembrane space) is always coupled with one reaction at the other side of the membrane. One reaction of reduction is coupled with a reduction of oxidation and vice-versa. The whole phenomenon involves transfers of electrons inside the membrane and transfers of protons from one side of the membrane to the other side. Note that some complexes belong to the membrane but only face one of its sides.

Q is the ubiquinone, a cofactor that belongs to the membrane. It interacts with the cytochrome c in a loop of oxidoreduction. One can see that the oxygen is involved by the complex IV. To resume the loop, we can write two reactions of oxidoreduction, one with the NAD and one with the FAD.

However O₂ cannot directly interact with FADH₂ and NADH+H⁺ and this loop is thus necessary. For redox, the free energy of Gibbs is

To determine the difference of potential ∆E°, we look at the couples involved in the reaction:

For FAD we have 43.3kcal/mol. There is a transfer of protons involved in the process but there must be an equilibrium between the concentrations of charge inside the cell and outside the cell. The protons can move out of the matrix of the mitochondrion through some protein complexes that generate ATP from the flow of protons (bottom of the figure). The F₁-ATPase protein is composed of two parts, F₀ in the membrane and F₁ in the matrix of the mitochondrion. F₀ pumps the protons from the intermembrane space towards the matrix. F₁ receives the energy from the transport and transforms ADP into ATP. Those species are charged negatively and require a transport through the membrane (top of the figure). One ADP can go in the matrix only if one ATP is moving out of the matrix. The inorganic phosphates P_i are also transported through the membrane.

The energy to form the ATP comes from the gradient of pH. It generates a proton motive force.

Where the indexes mat and ext respectively matrix and its exterior (the intermembrane space). The gradient of pH generates a difference of electric potential ΔV.

Combining the two effects, we have

Dividing this expression by nF, we obtain the expression of a variation of free energy ∆p associated to the transfer of one mole of protons (in volt).

The coefficient 2.303RT/nF of the gradient of pH has a value of 59mV at 25°C.

We can measure the pH and the difference of potential: ∆V=0.14V and ∆pH=-1.4.

It corresponds to a variation of free energy equal to

To form one mole of ATP, we need 7.3kcal. In the respiratory chain, there are 4 complexes that oxidise the NADH and the FAD.  They respectively form the equivalent of 11 and 8 protons by loop.

We can write a global equation of the “combustion” of a pyruvate during the citric acid cycle and the respiratory chain:To summarise, the oxidation of one NADH generates ~3 ATP and the oxidation of FAD generates ~2ATP.

The 15 ATP come from

• 4 NADH (one before the citric acid cycle and 3 during the cycle) à12ATP,
• from 1 FADH₂ à 2ATP
• and from 1 GTP»1 ATP.

To form one pyruvate, glycolysis forms 2 ATP and uses 2 NAD that are reduced into NADH+H⁺.

So considering 3 ATP by NADH, it corresponds to the production of 8 ATP. Added to the 15 ATP of each pyruvate (x2), we obtain 38ATP by glucose. In comparison to the combustion of the glucose (∆G°’=-686kcal/mol), it represents about 40% of the potential energy (38×7.3kcal/mol=277kcal/mol).

The NAD involved in the glycolysis and that produces a NADH is not at the same place (cytosol) that the NADH required in the cellular respiration (matrix of the mitochondrion). The NADH cannot enter freely into the mitochondrion because it is charged. There is a system of shuttle that may differ for different cells and that will modify the NADH to allow its passage through the membrane, then modify it again so it can be used there. In the brain and in muscles, it is a shuttle of glycophosphate that we will next explain. The shuttle is basically composed of two reactions: one in the cytosol and one in the matrix of the mitochondrion:

The shuttle can be represented like this:

If you remember, dihydroxyacetone-3P was part of the glycolysis (red rectangle in the following figure): it is one of the two trioses phosphate produced from fructose -1,6-diphosphate. The fact that this triose is involved in the shuttle does not really decrease the yield of the glycolysis significantly: the cell needs a given quantity of dihydroxyacetone for the shuttle but the molecules are regenerated in the matrix of mitochondria and can then return to the cytosol. So they just have to be produced once. The rest of the production is turned into glyceraldehyde 3P for the glycolysis.

The FADH₂ is oxidised during the respiration and the dihydroxyacetone can move back to the cytosol. The shuttle has a small cost: it oxidises one NADH into NAD in the cytosol while the reaction in the matrix involves FAD/FADH₂. In term of energy, it represents one ATP used because all the processes don’t take place at the same place.

The malate-aspartate shuttle

The same cofactors (NADH+H⁺/NAD⁺) are used so the yield does not change with this shuttle (38 ATP). The mechanism is based on the reducing power of the oxaloacetate.

Reduced into malate (1), it is transported in the matrix of the mitochondrion (2) through the malate-α-ketoglutarate transporter if one α-ketoglutarate is available to make the displacement in the opposite direction. The reverse reaction, the oxidation of the malate into the oxaloacetate is made in the matrix (3). Yet, the oxaloacetate cannot goes back out of the membrane. To do so, it is transformed into an amino acid, the aspartate, by a reaction of transamination (4):

It is done here by the aspartate transaminase with the use of the glutamate as amino acid. As a result, the oxaloacetate is transformed into aspartate that can move through the membrane through a transporter requiring the passage in the opposite direction of a glutamate (5). In the cytosol, it is turned back into the oxaloacetate with the exact same reaction (6) than at the other side of the membrane.

The post Chapter 7b : Glucose catabolism – respiratory chain appeared first on BORZUYA UNIVERSITY.

The first step is to choose the set of coordinates in which we will work. A molecule with M nuclei has 3M coordinates: (x_i,y_i,z_i) for each of the M atoms.

The laboratory axes system (LAS) is the first set of coordinates that we use. In this system we determine the position of the centre of mass of the molecule. Those 3 coordinates allow us to determine the translation of the molecule but not the rotation or the vibration modes: the centre of mass doesn’t move because of those modes.

To determine those modes, we need two other sets of coordinates: the LAS’ that is fixed to the centre of mass of the molecule and consequently is independent of the translation and the MAS: molecular axes system that is fixed to the molecule and turns with it. 2 angles 0≤θ≤π and 0≤Φ≤2π are used to locate the linear molecules and a third one 0≤χ≤2π is needed for the nonlinear molecules. Those 3 angles are called the angles of Euler and are

From the 3M coordinates, we used 5 (linear molecules) or 6 (non-linear molecules). The rest correspond to the modes of vibration of the molecule. In a diatomic molecule, there will be only one mode of vibration: M=2 and 5 coordinates are used to locate it.

To resume, the coordinates of one molecule are given by the vector

with 3M dimensions. The translation of the centre of mass is determined in the LAS

The condition here reflects the facts that the centre of mass does not move in the LAS’ referent. The MAS rotates with the molecule. To move into this referent, we apply the matrix S to the vector R_A’.

The matrix S defines the orientation of the axes (x’,y’,z’) of the LAS’ from the coordinates (x,y,z) of the LAS:

Small vibrations around the equilibrium R_A^eq are given by the vector dA

with the conditions of Eckart that

The first condition reflects that the centre of mass does not move because of the vibration and the second condition that there is no rotation in the MAS.

The kinetic energy of the molecule is in this notation

with M_A the mass of the nucleus A and Ṙ_A=dR_A/dt. Replacing Ṙ_A by its expression

We obtain (the red terms equal zero)

If Eckart is respected, then the interaction term Trot/vib can be neglected. We can thus approximate that the energies of rotation and of vibration are separable. The order of magnitude is indeed different: the vibration is found into the infrared while the rotation is observed in the microwave range.

Vibration

For a diatomic molecule, the oscillation is characterised by a force of recall

We find that

The energy of vibration can thus be approximated to a constant. The kinetic energy T_vib is always positive and is thus equal to

From the quantum mechanics, we know that

The separation in energy between the levels characterised by a number of nodes v=0, 1, 2, … There is thus regular a ΔG_V=ῡ.

There is however a deviation to the harmonicity that we found. If we develop the potential as a series of Taylor, we obtain

The two first terms are simply equal to zero (V₀=0 and the potential is at a minimum at the equilibrium). The third term is the harmonic result that we just obtained but further terms express the deviation to the harmonicity. The potential of Morse gives an empiric formula that fits correctly the real potential.

The anharmonicity modifies slightly the energy of the states

In the case of polyatomic molecules, we can consider that all the nuclei oscillate in phase, giving a base of 3M-6(5) independent movements. The result doesn’t noticeably differ from the diatomic molecule in this case.

Rotation

The rotation can be considered as a rigid rotation: the difference of frequency and of energy between the rotation and the vibrations is huge enough to make this approximation. The angular speed ω is thus identical for all the nuclei of the molecule.

The kinetic energy due to the rotation for a diatomic molecule is thus

with I the moment of inertia that is common to the two atoms.

μ is the reduced mass of the molecule. The angular moment J is given by

Its absolute value is

For a diatomic molecule, we get

The angular moment and the kinetic energy are thus directly bound:

We can go from the classical mechanics to the quantum mechanics by the application of the Hamiltonian on a wave function.

The multiplicity is g_J=2J+1. A small correction has to be added due to the centrifugal distortion, correction which is normally very small and negligible except when the rotation is very fast:

The spectrum of the rotation is thus composed of bands regularly spaced.

The post Chapter 15 : MPC – Molecular degrees of freedom: vibration and rotation appeared first on BORZUYA UNIVERSITY.

The molecular orbitals have the symmetry of one of the irreducible representations of the group G. This symmetry is taken into account in the LCAO coefficients. Some are null (some symmetries are not used) and some are equals in absolute value: only functions of same symmetry interact together to form molecular orbitals.

Given a MO Φ_a ∈ D⁽ⁱ⁾ of G and the AO {χ}, we can adapt the atomic orbitals to the symmetry of D⁽ⁱ⁾: {χ}à{χ⁽ⁱ⁾}. Then

The advantage of doing this is that the number n⁽ⁱ⁾ of functions χ⁽ⁱ⁾ in the last expression is smaller than (or equal to) the number n of functions χ in the atomic orbitals. Let’s apply the LCAO theory to an example in which the orthonormal base is composed of two function {Ψ₁, Ψ₂}, i.e. a case where two states are in interaction with each other.

The secular determinant is

As the base is orthonormal, S is a delta of Dirac: S₁₂=S₂₁=0, S₁₁=S₂₂=1 and H₁₂=H₂₁. Posing that the two states H11 and H22 are equidistant from the zero energy, they are separated in energy by 2H:

We can represent the problem as follow:

The secular determinant is thus reduced to

and

If we set c₁=1, then c₂=(H+E)/H₁₂. The coefficients must still be normed.

Two cases can be considered:

• H=0 with H₁₂<0.

Two degenerated states interact with each other. The result of this interaction is that the states are repelling from each other. One is stabilised and the other one is destabilised by ΔE=H₁₂. We will obtain a bonding state and an antibonding state. H₁₂ is thus a measure of the interaction between the states. The bigger it is, the bigger is the separation between the resulting states. H was the distance between the states that interact together.

Posing that c₁=1, then c₂=-1. The coefficients must still be normed: c₁=1/√2 and c₂=-1/√2. As a result, the wave function of the state of energy E=-H₁₂ is

We can do the same for the second state (E=H₁₂) and obtain

• H≠0 and H>>H₁₂

The states that are interacting together have not the same energy. For instance, let’s consider H=2 and H₁₂=-1/2.

The interaction between the two states separated the states but just by a bit. The states did not mix a lot together.

The mixing between two states is inversely proportional to the difference of energy between the states.

If H₁₂=0, there is no interaction between the states. It is the case only when the states don’t share any symmetry, i.e. H₁₂ can be different from zero only if Ψ₁ and Ψ₂ have common proper values with all the operators that commute with Ĥ. As a result, a triplet does not interact with a singlet even if the energies of those states are similar.

Rule of non-cancellation of an integral:

An integral is not equal to zero if the integrand is invariant with regards to all the operation of the group G, i.e. if the reduction of the direct product contains the totally symmetric irreducible representation D⁽¹⁾. In other words, the integral is different from zero if the integrand is totally symmetric.

Cases of H₂⁺ and H₂

The orbitals 1s of the two atoms are interacting together to form molecular orbitals σ_g and σ_u. σ_g is a binding orbital resulting from a constructive interference:

Here we considered that the base is not orthonormal. It is why the norm is 1/√(2(1-S)) and not 1/√2. S is the deviation to the orthonormal base

In an orthonormal base, S=0. The antibonding state σ_u is resulting from a destructive interference

The interactions can be seen this way:

When the electron is in a molecular orbital such as the attraction it produces on both nuclei brings the nuclei closer to each other, the effect is binding.

In the figure above, the electron is between the two nuclei. The attraction is produces on the nuclei is represented by the black arrows. The green arrows are the projection of the attraction on the axis passing by the two nuclei. The nuclei move thus in the direction of the other nucleus and remain thus together because of the presence of the electron.

If the position of the electron leads to a separation of the nuclei, then the effect is antibonding. The interaction of the electron-nucleus is positive but the intensities and directions are such as the nucleus-nucleus distance increases.

In the picture above, both nuclei are attracted by the electron but they move in the same direction with different speeds. The nucleus of the right moves faster than the nucleus of the left and the nuclei move thus away from each other.

The function 1s can be expressed as an exponentially decreasing function centred on the nucleus.

We can superimpose the functions 1s of two hydrogen atoms. In the case of σ_g, the functions add together:

We can do the same for σ_u.

If we put those function to the square, we obtain the probability of presence of electrons.

 In the second case, there is a place between the nuclei where no electron can be found. No liaison can thus be done between the nuclei.

Interaction between 2p orbitals

All the 2p orbitals don’t interact the same way. The 2p_z orbitals interact together to give the σ orbitals. This time, it is not the sum of the atomic orbitals that give the molecular orbital of lower energy and the binding orbital σ_g.

The other 2p orbitals (2p_x, 2p_y) lead to the π orbitals.

Note that the energy of the orbitals depends on the atoms. They all decrease in energy with Z but not with the same speed (see the figure below).

The energy of the π_u orbitals is almost constant while σ_g 2p_x decreases quickly with Z. σ_g 2p_x falls under π_u at O₂.

The energy of the liaison between the two atoms increases up to N₂ and decreases after because electrons are placed in antibonding orbitals.

The Walsh diagram

This diagram relates the energies of molecular orbitals of a molecule as a function of the angle that separates the liaisons. It helps to visualise the stability of the liaisons with regards to the symmetry of the molecular orbitals. The following figure shows the Walsh diagram for AH₂.

On the left one sees the linear molecules. As we go towards the right, the angle between the two liaisons goes towards the right angle, i.e. towards a bent conformation. As the bond angle is distorted, the energy for each of the orbitals can be followed along the lines, allowing a quick approximation of molecular energy as a function of conformation. As we move towards the top, the energy of the liaisons increases. Note that the 1π_u orbitals are degenerated for an angle of 180° but separate if we change the conformation of the molecule.

For one molecule, we count the number of electrons of valence. For instance BeH₂ has 6 electrons of valence (4 for Be and 1 for each H). We place 2 electrons by line, starting from the bottom (note that the 1a₁ line, binding the σ_g orbitals, is not plotted. The last electrons are on the 1b₂ line. One can see that the most stable angle for this molecule is 180°. For BH₂ and CH₂, the molecules are bent. If one electron is excited, then the conformation of the molecule can change.

Method of Hückel

This method limits the LCAO method to the π electrons. The reason is that a lot of physicochemical properties of the molecules can be explained by the π-π* orbitals. In the method of Hartree-Fock, the secular determinant was

We have to solve the determinant for all the orbitals of the molecule, what can quickly become complicated. If we apply the method of Hückel on C₂H₄ for instance, there is only one π liaison in the molecule and thus only one secular determinant to solve. Considering two degenerated levels of energy α, the secular determinant is

α=H₁₁=H₂₂ is the energy of the perpendicular to the plane atomic orbitals of the carbons and beta is the energy of resonance/interaction.

The solution found with this method (we won’t do it here) is close to the one obtained with the Hartree-Fock method that consider all of the electrons.

This method can be extended to other systems with π electrons if we pose that

If we consider the butadiene, we consider it as the interaction of two π systems

With the increase of π electrons, there are more binding states and one can see that, looking from the bottom to the top, the organisation of the orbitals follows a simple rule: the number of times that the signs are reversed increases by one at each orbital. One talk about the “wavenumber” of the orbitals. Indeed, on the lowest energy state, all the orbitals are aligned. There is no change of orientation of the orbital. On the second lowest state, the two orientations of the orbitals are present but they are grouped. There is only one change of orientation. The third level has 2 changes of orientations, there are 3 changes on the fourth lowest state, 4 on the fifth, etc… If we look at the orbitals as a wave, the wavenumber is indeed increasing with the energy of the state.

When the amount of π liaisons increases,

• the separation in energy between the states of same type (binding or antibonding) decreases and for a large amount of liaisons (in polymer for instance), we talk about a band of valence for the block of binding states and about a band of conduction for the block of antibonding states, separated by a gap.
• the amount of states increases in both bands,
• the separation in energy between the band of valence and of conduction decreases.

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From the wave function we want to determine the energy of the orbitals but we can’t solve the equation exactly except for systems with one electron. For other species, we can only find an approached solution. To obtain it, we use the theory of perturbations or the method of variations.

Theory of perturbations

We can apply this theory to states that are independent of the time, i.e. stationary states. We can approximate the Hamiltonian and the energy of one state if this state is the result of a small perturbation λ of a state the solutions of which are known (Ĥ⁽⁰⁾, E⁰).

The solutions are then expressed as a series of correction to the model at order zero.

The corrections are calculated from the solutions at the order 0. For instance, the correction of order 1 are

The corrections usually decrease in intensity with their order but they will always approach the approximation from the exact solution. Note that we can go beneath the exact solution.

Method of variations

To find the exact energy, we use a trial wavefunction. This function has a known form but will probably not give the exact energy but will give us a superior born for the exact energy. Next, we modify the parameters of the trial wavefunction to obtain a better approximation of the energy. Unlike the method of perturbation, we can’t go beneath the exact energy with the method of variations. Each time we modify the parameters, we obtain a solution of lower energy and we approach the exact value of the energy. For instance if we chose a wavefunction with two parameters α and β. We can try to improve the values of the parameters to obtain the best possible approximation.

We reach an optimal estimation when the energy does not vary anymore when we change the parameters, i.e. when dE/dparameters=0.

Application to the helium

We will apply those two methods to estimate the energy of the helium. The exact solution is E=-2.903u.a. and the complete Hamiltonian is

One model the solution of which is known is the hydrogenous model:

The solution of this model for 1 electron is

with Z the atomic mass and the quantic number n. As there are two electrons in the helium, we simply multiply the energy by 2 to obtain a first approximation from the hydrogenous model:

This approximation is far from the exact solution and overestimates the stability of the atom because we did not take the repulsion between electrons into account (the +1/r₁₂ term).

The theory of the perturbation will start from this model and introduce the repulsion term as a pertubation of the model:

This estimation is way better than the hydrogenous model alone.

The method of variations is slightly different as we choose a wave function Ψ that depends on a few parameters that we may vary. As for the theory of perturbations, we select the hydrogenous wave function

As there are two electrons in the helium, we use a combination of two wave functions:

The parameter that we will vary is the effective charge Z=Z_eff. One part of the charge of the nucleus is indeed hidden to one electron by the second electron.

From the chosen wave function, we find an equation for the energy that depends on Z

We optimise the parameter to obtain the best approximation that we can, such as dE/dZ=0. Solving this, we find

We apply this particular value of Z to the equation for the energy to find

This result is close to the exact solution.

Principle of linear variation: linear combination of atomic orbitals (LCAO)

In this method, we assume that Ж can be expressed as a linear combination of a base of functions {Ψ₁, Ψ₂, …}

where c_i is the variational parameter associated to the wave function Ψ_i. These coefficients are adjusted to approximate the exact energy. For more simplicity, let’s take an example in which Ж is a linear combination of two wave functions Ψ₁ and Ψ₂.

It is corresponding to the case of H₂:

The Hamiltonian is

Next we multiply this equation by Ψ₁* and integer it over tau:

We can do the same with Ψ₂*:

Now, we introduce H_ij and S_ij such as

The previous integrals can be written

or

This system of two equations can be written as a 2×2 secular determinant

In a general way, it forms a n by n secular determinant if there are n wave functions (or particles)

The n solutions of the secular determinant give the n most stable states of energy of the system. If we inject one of the energies E=E_q in the system, we obtain the optimised coefficients {c_iq} related to the state q.

If the base is orthonormal, then

And then

If the base is complete, the linear combination corresponds to the exact solution due to the theorem of superposition (Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together (“superposed”) and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states). The more the base is complete, the more one tends to the exact solution.

Method of Hartee-Fock

This method is an application of the variational method in which the trial wave function is a normalised determinant of Slater Ж = ∣ Φ ₁ Φ ₂ Φ ₃… Φ _N ∣. We try to minimise the energy of the orbitals (∂E/∂Φ_i=0) and to keep them orthonormal. We obtain a system of N coupled equations of Hartree-Fock that describe the movement of one electron in the averaged field u_b created by the other electrons, such as

where ε_a is the HF energy of the molecular orbital Ψ_a. If we introduce the operator of Fock

we can rewrite the system of N equations as

or on one line:

This equation only depends on the position and the movement of one electron, the effect of the other electrons being averaged. As the averaged field is determined by the orbitals that we are looking for, we have to solve the system of equations by iterations, starting from a set of orbitals of trial.

The convergence is guaranteed by the variational principle. In practice, we develop each orbital as a linear combination of Gaussian atomic orbitals centred on the nuclei. Then we iterate.

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M_L is the quantic number associated to the projection of L on the internuclear axis.

The projection is degenerated because it can either be in the positive values of the z axis or in the negative ones. The projection of L can thus give M_L or –M_L. We can define a new quantic number L=∣M_L∣.

The fact that there are two projections explains the dimension 2 or the orbitals π, δ, … of linear molecules. The length of the projection of L grow if  L gets larger but the degeneration is always 2. The only exception is Σ for which M_L=0. In the atoms we had the possibility to choose the orientation but we cannot do that with molecules.

As a result, the operator σ_v doesn’t commute with L_z:

except for L=0 that gives the states Σ⁺ and Σ ^–. This distinction + or – is not present for the states Π, Δ, …

The fact that σ_v doesn’t commute with L_z induces the degeneration of the states.

The inversion operator Î still commutes with Ĥ, L_z and σ_v in the case of centrosymmetric molecules.

As a resume, the operator σ_v is separated from the other operators of the CSCO of linear molecules because it does not commute with L_z anymore.

Application to H₂

Let’s take a look at the possible electronic configurations of the molecule H₂.

The unexcited state of H₂ has two equivalent electrons on the ground orbital 1σ_g:

This state is binding: between the nuclei, the probability of presence of electrons is positive. In antibonding states, there are some points between the nuclei where the probability to find electrons is zero.

The first excited state is the configuration 1σ_g1σ_u. In this configuration the electrons are not equivalents and the degree of degeneration is 4 (2×2):

To determine the states, we take the emergent (M_L=0,M_S=1). It corresponds to the triplet ³Σ _u ⁺ (the pairs (0,1), (0,0) and (0,-1)). The second emergent (0,0) corresponds to the singlet state ¹Σ _u ⁺(the pair (0,0)).

To determine the energy of the states, we proceed as for the atoms with the determinants of Slater

In the case of an emergent such as (0,1) we have a proper function and thus the energy can be determined. From this function we find the other functions with operators of rise/descent S₊ and S_– and with the principle of orthogonality we find the energy of the singlet state.

The dashed curves are correspond to unstable states because there is no minimum of energy: the atoms are get more stable as they get away one from each other.

Application to O₂

The same method can be applied to O₂. Its fundamental configuration is

As usual, we only consider the highest occupied molecular orbitals (HOMO). Those are the 2 1π_g orbitals with 2 electrons to place, represented above by the circles: they can be on the same orbital or separated. There is thus a degeneration of 6:

The first emergent is (2,0). That corresponds to the group 1Δ_g. Remember that the degeneration for linear molecules is 2 except for M_L=0. In the atomic case, we had a degeneration of 2L+1 (i.e. M_L, M_L-1, …, 0, …, –M_L-1, -M_L) but here we just have M_L and –M_L. The next emergent is (0,1), a triplet ³Σ_g. Finally there is a singlet ¹Σ_g. To know if it is Σ⁺ or Σ^–, we have to apply the operator σ_v. In this case we have ³Σ_g^– and ¹Σ_g⁺.

Ozone is obtained by the excitation of one molecule of O₂ at its fundamental state into two oxygen atoms in the ³P state. In this state, they react with another molecule of O₂ and a catalyst to produce the ozone.

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When two operators of the CSOCH don’t commute together, it implies a degeneration of the states.

In the case of atoms, when we have a spherical symmetry (it has been shown recently that some atoms are slightly deformed and have an elliptic shape (and then a quadrupole moment) or a pear shape (and then an octupole moment)), we have the usual CSCO. We can build 3 SCO:

This last group is present for any system of electrons: the spin of the electrons is independent of the geometry of the molecule. The small letters correspond to operators of orbitals. In the case of a linear molecule, we lose the spherical symmetry but there is still a cylindrical symmetry. Instead of an infinity of axis of rotation passing by the nucleus, we have now only one axis of rotation on the axis of the two (or more) nuclei. This difference induces a modification of the SCO’s: we lost the L² symmetry. As reminder, L²=L_x² + L_y² + L_z². L_Z is still in the SCO but L_y and L_x are not anymore commuting with H. We have now

In the case of nonlinear molecules, we also lose the operator L_Z.

We have other symmetry operators that we have to add to the CSOCH (showed by the ? above):

• a bilateral symmetry σ_v: there is an infinity of planes of symmetry passing by the axis of the molecule.

• a centre of inversion Î if the linear molecule is centro-symmetric (same atoms at each side of the centre of the molecule, ex: CO₂, N₂, C₂H₂).

Those operators are necessary to describe completely the molecule: the operators of symmetry characterise the spatial behaviour of the system. As a result, the ensemble of all the operators that commute with H define the state of the system from the spatial point of view. If we forget some operators in the CSOCH, one part of the quantic information is lost. We say that the ensemble of the operations of symmetry form a mathematic group.

Theory of groups

 G={a, b, c, …} forms a mathematic group with regards to one law (*) if

• * is intern and defined everywhere: a*b=c a, b, c ϵ G
• * is associative: (a*b)*c=a*(b*c)
• ∃ e neutral (e ϵ G) : a*e=e*a=a
• Reversibility: ∀ a, ∃ x=a^-1 ϵ G : a*a^-1=e.

For a group of symmetry, a*b means that we apply the operation b first then the operation a.

The order h of a group of symmetry G is the amount of operations it contains. A group of symmetry can be continuous (order h of G is infinite, i.e. a symmetry of revolution) or finished (h is finite), commutative (or abelian) if a*b=b*a ∀ a and b ϵ G, or non-abelian (what leads to a degeneration).

The representation of a group is a set of n by n matrices (n being the dimension of the representation) {D(a), D(b),…} associated to the elements {a, b, …} of G such as

• the matrix product is associated to the law (*)
• the matrix unity is associated to the neutral e.

The representation is said to be reducible if the matrices can be diagonalised and irreducible if it is not the case. Any reducible representation D of G can be expressed as a linear combination of the irreducible representations D_i of G.

We can thus search for the operators of symmetry of a molecule that let the Hamiltonian unchanged. It is the interactions electron-nucleus that impose this symmetry.

There are 5 operations of symmetry:

• identity: E – no displacement
• inversion: I – central symmetry or centre of inversion

• reflexion: σ_v(vertical) σ_h(horizontal), σ_d(diedre) – planar or bilateral symmetry

• proper rotation: C_n – rotation of 2π/n rad around the axis

• improper rotation: S_n – commutative product of a rotation of 2π/n around the axis with a reflexion in the plane perpendicular to the axis.

To apply several consecutive rotations we add an exponent x to C_n or S_n. It means that we apply the rotation 2π/n x times. x goes from 0 to n for Cn and up to 2n for Sn. The Some elements are equivalent. For instance C³₃=E

The groups are named following this picture:

Now lets come back quickly on the properties of the groups.

Faire pareil: d’abord expliquer avec les opérations sur la base 3 puis dire que pour l’explication il était plus simple de montrer avec les axes x,y,z mais qu’en réalité, en utilisant  d’autres coordonnées on arrive à une représentation de dimension 4 qui donne lieu à une ligne supplémentaire.

Let’s take a look on how we build the representation of the group of symmetry C_2V, i.e. the group of shouldered molecules as H₂O, NO₂ but also as CH₂O. If we draw H₂0 in the Cartesian coordinates such as

If we apply an operation of symmetry on the molecule, for instance σ_v(xz), i.e. the reflexion in the plane xz, the molecule did not change but if we follow one of the hydrogen atoms, its coordinate y changed of sign ((x,y,z)à(x,-y,z)). We can translate this into a matrix of dimension 3 that we apply on the coordinates x, y and z:

Such a matrix can be found for the other operators of the group

These matrices commute together and the group is intern and defined everywhere. For instance

Each column of the representation of the group is the value on the diagonal of the matrix for the corresponding base (here the coordinates x, y, z)

Those values are associated with the quantic numbers and the parity. 1 means symmetric and -1 antisymmetric. The character of one matrix associated to the operation a is the sum of the elements on its diagonal and is noted χ(a).

Let’s come back to something we said earlier: any reducible representation D of G can be expressed as a linear combination of the irreducible representations D_i of G.

The coefficients can be determined from the characters of the matrices associated to the operations of this group because of the properties of orthogonality.

Where χ ̅(a) is the value of the irreducible representation associated to one base. For instance, in the base A₁ we have χ ̅(E)=1, χ ̅(C₂(z))=-1, χ ̅(σ_v(xz))=1, χ ̅(σ_v(yz))=-1

Applied to the C_2v group, it gives

We will see next that there is in fact a fourth line (A₂) in the table and why it is necessary to correctly describe molecule.

And thus

A group is normed if its order h, equal to the amount of operations, is also equal to the sum of the squares of χ ̅(a), and this for each irreducible representation, i.e.

For C_2v, h=4=1²+1²+1²+1² (in the case of the operation E). For the operation C₂(z) we find the same value: h=4=1²+1²+(-1)²+(-1)². Two representations k and l are orthogonal if

For instance, we obtain for A₁ and B₂

The group C_3v describes molecules such as NH₃. It contains 6 operations: E, two C₃ and three σ_v. We can represent it as

The irreducible representation E is degenerated: it contains a value different from -1,0 or 1. Yet the group is still normed (don’t forget there are two C₃ and three σ_v):

The representations are orthogonal (for instance between A₁ and E):

Application of the theory of groups

Each orbital of the atoms of one molecule is a basic function that impacts the geometry of the molecule. We need all of those functions to describe correctly and completely the molecule. As a result, a molecule such as NO₂ is described by one representation of base 15 that can be reduced into 10 complete bases (5 of dimension 1 for the nitrogen and 5 of dimension 2 for the oxygen) because the two oxygen atoms are equivalent.

We have seen previously that the NO₂ molecule belongs to the group of symmetry C_2v. If one representations of N can be transformed by one operator of this group into itself in absolute value, it means that this representation is an irreducible representation of the group for the atom N. It is indeed the case for NO₂:

• all of the orbitals of N are irreducible: the orbitals s are spherical so the operation of symmetry changes nothing while the orbitals p change of sign for some operations of symmetry.

• the orbitals of the oxygens are transformed into themselves or into an orbital of the other oxygen, in absolute value.

We can thus find a matrix D(R) such as when it is applied to the base we find the linear combination that belongs to the group.To build a representation of the group C_2v, we have to build one matrix of dimension n for each operation of symmetry of the group, i.e. 4 matrices. Those matrices define the transformation with regard to the four operations of symmetry of one base of n functions. The base has to be complete, i.e. the action of one operation on one of the functions of the base has to give a linear combination of functions of the base. In mathematical words,

The set {D(R), D(R’),…} is a representation of the group G.

The trace T(R) is the sum of the diagonal members of the matrix D(R) (here it is a+d). The set {T(R), T(R’),…} characterises the representation of the group D.

From a few irreducible representations of one group, we can find the other representations of the group. We proceed that way:

3 representations can be found from the nitrogen (see the table behind).  One representation (A₁) is when none of the orbital changes during any operation of symmetry. This representation works for the orbitals 1s, 2s or 2p_z. A second representation (B₁) stands for the orbital 2p_x for which C₂(z) and σ_v(yz) change the sign of the function. The third representation (B₂) is when C₂(z) and σ_v(xz) change the sign of the function, i.e. in the case of the orbital 2p_y.

We can write the representations in a table

These three representations are not the only ones that apply to the NO₂ molecule. We must also describe the orbitals of the atoms of oxygen to give a complete description of the molecule. In this case, it is a bit more complicated because the operations of symmetry can transform one oxygen into itself or into the other oxygen, in absolute value. As a result, the representation is not a series of numbers but a series of 2×2 matrices. If the orbitals remains unchanged by an operation, the matrix corresponding to this operation of symmetry is

If the atoms are interchanged, the matrix is

and if the sign changes it is

For instance, D(E) of the 1s orbitals is (1 0 sur 0 1) because

The operator σ_v(xz) interchanges the atoms of oxygen: and D(σ_v(xz))=(0 1 sur 1 0) because

And we can do that for the other operations as well. That being done, the table of the group C_2v is completed:

However, there is still a problem: the representations that are matrices are not irreducible. But we can find from which linear combination of irreducible representations they come from the traces of the matrices.

One can see that we can combine the representation A₁≡{1 1 1 1} with the representation B₂≡{1 -1 -1 1}. The representation corresponding to the base (1s₁, 1s₂) is thus A₁ + B₂. We can repeat this process for the other bases of the group. It works fine for 3 bases but not for the base (2p_x1, 2p_x2). There is thus an unknown representation A₂ in the group so that we can build the matrices of the base (2p_x1, 2p_x2). Obviously the combination will involve the representation B₁ to obtain a trace of -2 for the operation σ_v(yz). The representation A₂ is thus A₂≡{1 1 -1 -1}.

If you remember well, earlier in the course we discussed on the operators of symmetry of the water and we already talked about a fourth representation in the group C_2v. Now you know why we needed this fourth representation. It is required to build the CSCO that would not be intern without A₂.

Amongst the 15 atomic orbitals of NO₂, we have 7A₁ + 1A₂ + 2B₁ + 5B₂:

7A₁: 1s, 2s, 2p, (1s₁,1s₂), (2s₁, 2s₂), (2p_y1, 2p_y2), (2p_z1, 2p_z2)

1A₂: (2p_x1, 2p_x2)

2B₁: 2p_x, (2p_x1, 2p_x2)

5B₂: 2p_y, (1s₁,1s₂), (2s₁, 2s₂), (2p_y1, 2p_y2), (2p_z1, 2p_z2)

To build the representation A₁, we will thus need a linear combination of the 7 orbitals.

Method of projection

The projection allows to determine which orbitals take part to which representation. In a general way, the operator of projection P⁽ⁱ⁾ for  non-degenerated representations is given by

with i the corresponding representation, χ the trace of the operator and h the norm. For instance, applied to NO₂,

To obtain π liaisons, the p orbitals have to be oriented correctly. On the other hand, antibonding liaisons are the result of p orbitals that are (also) correctly oriented but with opposed signs. We know that there are 3 unoccupied orbitals in NO₂ (15 orbitals and 23 electrons). These orbitals are some of the antibonding orbitals and are the highest in energy.

For degenerated representations, we can still do the projection using the diagonal elements of the matrix D(i). The formula changes a bit:

with n_i the degree of degeneration.

Application to NH₃

NH₃ belongs to the C_3v group.

Obviously the presence of a 2 in the irreducible representation E indicates that this IR is degenerated. The orbitals to consider are

The orbitals s of N belong to A₁ because they are spherical and the 2p_z also does because it is on the axis of rotation. The rotations that we can apply on NH₃ are of 120°, so our system of coordinates is not very convenient here. We can however obtain the matrix representation of {2p_x, 2p_y}

The matrix representation of {2p_x, 2p_y} is thus the irreducible representation E: the traces of the matrices correspond to the elements of the IR E in the table of C_3v.

We repeat the process for the three hydrogen atoms

It doesn’t correspond to any IR of the table C_3v but we can find a linear combination of them:

There are two projections for the operation E as it is degenerated twice. The atomic orbitals adapted to the C₃ symmetry are, for the hydrogen atoms

We already verified the orthogonality of C_3v previously

Proper functions

The proper functions of Ĥ (polyelectronic states) and of ĥ (molecular orbitals) that have a proper value E in common form a base for the irreducible representation of G.

Non-degenerated case:

We can thus reduce the molecular representation of dimension n into n matrix representations of dimension 1.

Degenerated case (n times):

Several proper functions have the same energy. As a result, the reduction leads to several representations amongst which some are not of dimension 1.

Direct Product of two representations

The direct product between two representations of dimensions n and m give a n.m representation.  It allows to determine the symmetry of the product of two or more representations, i.e. in case of coupling between orbitals. For instance, the direct product of one representation D⁽¹⁾ of base f (dimension n₁=3)  with one representation D⁽²⁾ of base g (dimension n₂=2) gives a representation D of base fg (dimension n=n₁.n₂=6)

The character of the new representation will simply be the product of the characters of the old representations (no need to do all the stuff).

We can thus determine the symmetry of the product of several functions.

The direct product with one 1D representation gives an irreducible representation. If the representation is reducible, it is a linear combination of the representations of the group.

Most of the molecules have a fundamental state that is an A₁ representation because the octet rule is respected. I understand that the direct product and its interest are vague for now but we will see them with the example of NH₃.

 Example of NH₃

NH₃ belongs to the group of symmetry C_3v.

The direct product of two E representations gives in the group C_3v gives

This product corresponds to a linear combination of the IR of the group.

The coefficients can be determined from the relation

The molecular orbitals are called in function of their irreducible representation, written in small letters. The fundamental state of NH₃ is

The MO 1a₁² results from a coupling between two A₁ states: A₁xA₁=A₁. It is the same for the other a₁² molecular orbitals. For the 1e⁴, we have the coupling of 4 E representations. The E representation is of dimension 2 and the coupling gives a degeneration of 16 (2⁴). However, the principle of Pauli has to be respected as well, i.e. the only possibility is that the 4 electrons are distributed amongst the two orbitals of same energy with opposite spins. As the layer is full, the representation is A₁.

If one electron of the HOMO (orbital 1e⁴) is excited to the orbital 2e (LUMO) we obtain two ²E states (1e³ and 2e¹ give the same state by the symmetry hole/particle).

Cases of the spherical and cylindrical molecules (or atoms) – group of symmetry K_h

I don’t know the exact reason, but for those two cases the representations have specific names. For atoms, the representations have names identical to the atomic symmetry. The orbitals s belong to the group S (dim 1), the orbitals p belong to the group P (dim 3), etc. We can put an index to the group that translates the parity in this group. The index is g (gerade from German) if the state is symmetric and u (ungerade) if the state is antisymmetric.

If we apply the inversion operator on the nitrogen, we have

The coupling between two identical indexes gives gerade and between two different indexes gives ungerade:

The groups for linear molecules that are not centrosymmetric are called differently

The distinction between σ⁺ and σ^– is the direction of the rotation (clockwise or anticlockwise).

For centrosymmetric molecules, we note the distinction between ungerade and gerade

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