A hard problem & a hard place

consciousnessConsciousness and physics: A hard problem leads to a hard place

If consciousness is related to quantum biology, the study of consciousness is going to require some concept of the underlying physics, however unpalatable this may be to some of us. We are dealing with a hard problem, and we will have to come to a hard place to solve it. The summary below attempts to provide a very simplified version of the necessary underlying physics drawn from conventional sources.

Atomic bonds:  The overlap of the atomic orbitals forms bonds between atoms, and thus creates molecules, and also determines the shape of a molecule. The same atoms held in a different shape can result in a different compound. The energy level of each orbital is known as ‘n’.. Each value of ‘n’ can represent a group of orbitals at different energy levels known as a shell. The first shell, n = 1, can only contain one orbital, the second shell, n = 2, can contain two orbitals, the third shell, n = 3, can contain three orbitals and so on. P.

Angular momentum:  Another quantum number ‘L’ relates to the angular momentum of an electron in an orbital. The value of ‘L’ is at least one less than the value of ‘n’. The values for ‘L’ are conventionally given by letters. For our purposes here we need only deal with the values of 0 and 1, which are labelled ‘s’ and ‘p’. So an electron can be labelled 2s, denoting an orbital energy of 2 and an angular momentum of 0, or it can be labelled 2p with an orbital energy of 2 and an angular momentum of 1.

Electron wave function:  The electron orbital is viewed as being a wave function. With a wave, the wavelength or frequency is related to the energy level of the individual quanta, but the amplitude (the height of the wave) squared is the strength of the signal, or in other words the number of quanta involved . With a photon, the quanta of light, frequency determines the colour of visible light, but the square of the amplitude, signifying the number of quanta determines the brightness.

It is possible to chart the probability of an electron being present at a particular point in space, and this can be referred to as a density plot. For an ‘s’ orbital the density plot is spherical, but with ‘p’ electrons, the shape of the density plot is two lobes with a nodal area in between, where there is no electron density. The wave functions of these two lobes are out-of-phase.

A further quantum number mL relates to the spatial orientation of the orbital angular momentum. This gives a value of L- or L+ for ‘p’ orbitals, while ‘s’ orbitals have a 0 because a sphere does not have orientation in space. For ‘p’ orbitals there are three possibilities of -1, 0 and +1 that can be related to the mutually perpendicular x, y, and z axes in geometry, and are written as px, py and pz.

Structure of an atom:  The structure of an atom involves having two electrons in the lowest energy orbital and working up from there. Hydrogen has one electron located in the lowest energy orbital, and helium has two electrons placed in this orbital. Two electrons render an orbital full. An orbital can be full (2 electrons), half-full (one electron) or empty. With lithium which has three electrons, the third electron has to be located in a second orbital. With carbon there are six electrons, with two in the ‘n’ = 1, first shell. In the second, ‘n’ = 2 shell,  there is one full orbital with two ‘s’ electrons and two half-full orbitals each with one ‘p’ electron.

Structure of molecules:  The structure of the individual atom is also the basis for the structure of molecules. Atomic orbitals are wave functions, and the orbital wave functions of different atoms are like waves, in that if they are in phase, their amplitudes are added together. When this happens, the increased amplitude of the wave function works against the mutual repulsion of the positively charged atomic nuclei of different atoms, and works to bond the atoms together. This is referred to as a bonding molecular orbital. When the orbitals are out-of-phase, they are on the far sides of the atomic nuclei, which continue to repel one another due to like positive electric charges, and this arrangement is known as the anti-bonding molecular orbital. Collectively the two types of molecular orbital are referred to as MOs. The antibonding MOs usually have higher energy than the bonding MOs. Energy applied to an atom can promote a low-energy bonding orbital to a higher-energy anti-bonding orbital, and this process can break the bond between two atoms. When ‘s’ orbitals combine, the MOs are symmetrical, and this type of orbital overlap has sigma (σ) symmetry, and is described as a sigma (σ) bond.

When there is a combination of p orbitals, there is a possibility of three different p orbitals on axes that are perpendicular to one another. One of these can overlap end-on with an orbital in another atom, and these two orbitals are described as 2pσ and 2pσ*. Two other orbitals will overlap with those on other atoms side-on, and will not be symmetrical about the nuclear axis. These are described as π orbitals and form π bonds.

In discussing bonding, only the electrons in the outermost shell of the atoms are usually relevant. For example, in a nitrogen molecule formed by the bonding of two nitrogen atoms, only the electrons in the second, ‘n’ = 2, shell are involved in bonding. The nitrogen atom has seven electrons, so there are fourteen on the two atoms that bond to form a nitrogen molecule. Two electrons in the inner shell of each atom are not involved, leaving five on each atom and ten altogether in the second shells. The 2s electrons on each atom cancel out, and are described as lone pairs. The bonding work thus devolves on three electrons in each atom, or six in the whole molecule. These form one σ bond and two π bonds. This is described as a triple-bonded structure. Orbitals overlap better when they are in the same shell of their respective atoms. So electrons in the second shell will overlap more readily with other second shell electrons than with third or fourth shell electrons. Further to that p electrons must have the right orientation and px electrons can only interact with other px electrons and so on, because the x, y and z electrons are perpendicular or orthogonal to one another.

Molecular bonding also applies to molecules that are formed out of different types of atoms, as distinct from molecules formed from atoms of the same element such as the nitrogen molecule. If the atomic orbitals of different atoms are very different, they cannot combine, and the atom cannot form covalent bonds (sharing the electron between two atoms). Instead an electron can transfer from one atom to another, transforming the first atom into a negative ion, and the second atom into a positive ion, with the molecule now held together by the attraction between the oppositely charged ions. This is known as ionic bonding. Covalent bonds with overlapping orbitals can only be formed when the difference in energy is not too great.

Hybridisation:  Hybridisation is an important factor in the formation of molecular bonds. The ‘s’ and ‘p’ orbitals are those most important for organic chemistry and for the bonding of atoms such carbon, oxygen, nitrogen, sulphur and phosphorous. Hybridised orbitals are viewed as ‘s’ and ‘p’ orbitals superimposed on one another.

In its ground state, the carbon atom has two electrons in the first shell, and this is not normally involved in bonding. In its second and outer shell it has two ‘s’ electrons filling an orbital, and two ‘p’ electrons, one px and one py, each in a half-filled orbital. If the carbon atom is excited, say by the positive charge attraction of the nucleus of a nearby hydrogen atom, an ‘s’ electron in the outer shell can be excited into a ‘p’ orbital, so that the outer shell now has one ‘s’ electron and three ‘p’ electrons, one each in an x, y and z orientation. The four outer shell electrons are now deemed to be not distinct ‘s’ and ‘p’ electrons but four ‘sp’ electrons, here described as sp3, because the configuration is one quarter ‘s’ electron and three-quarters ‘p’ electrons. The arrangement allows the formation of four σ covalent bonds. Carbon atoms can use sp2 hybridisation where one ‘s’ electron and two ‘p’ electrons in the outer shell are hybridised. There is also ‘sp’ hybridisation where the ‘s’ orbital mixes with just one of the ‘p’ orbitals.

With the C = O double bond, the two atoms in the double bond are sp2 hybridised. The carbon atom uses all three orbitals in the sp2 arrangement to form σ bonds with other orbitals, but the oxygen atoms use only one of these. In addition a ‘p’ electron form each atom forms a π bond.

Delocalisation and conjugation: The joining together or conjugation of double bonds is important for organic structures.  π bonds can form into a framework over a large number of atoms, and are seen to account for the stability of some compounds. The structure of benzene is relevant in this respect. Benzene is based on a ring of six carbon atoms. The carbon atoms are sp2 hybridised, leaving one ‘p’ electron per carbon atom free, or six electrons altogether. These six electrons are spread equally over the six carbon atoms of the ring, a behaviour sometimes referred to as resonance. These are π bonds delocalised over all six atoms in the carbon ring, rather than being localised in particular double bonds.

Delocalisation can also be referred to as resonance. Delocalisation emphasises the spatial spread of the electron waves, and occurs over the whole of the conjugated system. Sequences of double and single bonds also occur as chains rather than rings. Conjugation refers to the sequence of single and double bonds that form either a ring or a chain. Double bonds between carbon and oxygen can be conjugated in the same way as double bonds between carbon atoms. Conjugation involves there being only one single bond between each double bond. Two double bonds together also do not involve conjugation. These ‘rules’ relate to the need to have ‘p’ orbitals available to delocalise over the system.

In both rings and chains every carbon atom is sp2 hybridised leaving a third ‘p’ electron to overlap with its neighbours, and form an uninterrupted chain. The double bonds that are conjugated with single bonds are seen to have different properties from double bonds not arranged in this way. Here again conjugation leads to a significantly different chemical behaviour.

Chlorophyll, the pigment molecule in plants, is a good example of a conjugated ring of single and double bonds, and the colour of all pigments and dyes depends on conjugation. The colour involved depends on the length of the conjugated chain. Each bond increases the wavelength of the light absorbed. With less than eight bonds light is absorbed in the ultra-violet.

An important feature of benzene is the ability to preserve its ring structure through a variety of chemical reactions. Benzene and other compounds that have this property are termed aromatic. In looking at these structures, the important feature is not the number of conjugated atoms, but the number of electrons involved in the π system The six π electrons of  benzene leave all its molecular orbitals fully occupied in a closed shell, and account for its stability. A closed shell of electrons in bonding orbitals is a definition of aromacity.

Delocalisation and conjugation

The colours of objects and materials around us are a function of the interaction of light with pigments. Pigments are characterised by having a large number of double bonds between atoms. The pigment, lycopene, responsible for the red in tomatoes and some berries, comprises a long chain of alternating double and single bonds, allowing the molecule to form π bonds. An extensive network of π bonds across a large number of atoms is involved in the chemistry of many compounds. It is responsible for the high degree of stability in aromatic compounds including benzene.

The compound ethylene (CH2=CH2) has all its atoms in the same plane, and is therefore described as planar. In this molecule, the two carbon atoms are joined by a double bond. Hybridisation involves mixing the 2s orbital on each carbon atom with two out of the three ‘p’ orbital on each carbon atom to give three sp2 orbitals. The third ‘p’ orbital on each atom overlaps with the ‘p’ orbital of the other atom to form a π bond. The ‘p’ orbitals of the two atoms also have to be parallel to one another in order to form a π bond. This bond prevents the rotation of the double bond between the carbon atoms. However, sufficient energy, such as that of ultra violet light, can break the π bond, and thus allow the double bond to rotate.

In benzene, the lowest energy ‘p’ orbitals comprise electron density above and below the plane of the molecule. These electron orbitals are spread over, delocalised over or conjugated over all six carbon molecules in the benzene ring. The delocalised ‘p’ orbitals can themselves be thought of as a ring. Expressed another way, this type of delocalisation is an uninterrupted sequence of double and single bonds, and it is this which is described as conjugation. The properties of this type of system are seen to be different from its component parts.

Benzene has six π electrons, and in consequence all its bonding orbitals are full, giving the molecule a closed structure, which is often not the case for quite similar molecules with a lot of double bonds. This is referred to as a molecule being aromatic. The general rule is that there has to be a low energy bonding orbital with the ‘p’ orbitals in-phase. There is a closed shell giving greater stability in aromatic systems, where there are two ‘p’ orbitals forming a π bond and four other electrons.

It is not essential in these systems to have carbon-to-carbon bonds. Carbon and oxygen also often form double bonds, separated by just one single bond. Here to the behaviour of the double-bonded system is quite different from the behaviour of the component parts. These structures are special in the sense of only arising where there are ‘p’ orbitals on different atoms available to overlap with one another. In many other molecules, there is a similarity in terms of a large number of double bonds, but they are insulated from one another by the lack of ‘p’ orbitals available to overlap with one another.

The amide group is crucial to protein, and therefore to living systems as a whole, in that it forms the links between amino acid molecules that in turn make up protein, the basic building blocks of life. The amino group on one amino acid molecule combines with the carboxylic group on another amino acid molecule to give an amide group. When a chain of this kind forms it is a peptide or polypeptide and longer chains are closed as proteins. Conjugation arises from the bonding of a lone pair of ‘p’ orbitals, and this is vital in stabilising the link between the amino acids, and making it relatively difficult to disrupt the amino acid chains that make up protein.

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